FORMULAS  IN 


ENGIN. 
LIBRARY 


B    3    112    3SM 


00 
00 

o 


LIBRARY 

t 

OF   THE 

UNIVERSITY  OF  CALIFORNIA. 

Deceived 

Accessions  No.  (&U410/  .  Class  No. 
tidal 

'Library 


5  /5ZZv  -/  ^-v^u^JLu*    (L 

M 


FORMULAS 


IN 


GEARING. 


WITH    PRACTICAL    SUGGESTIONS. 


UHI7BRSIT7 


PROVIDENCE,  R.  I. 

BROWN    <fe    SHARPE    MANUFACTURING    COMPANY. 

1892. 


, 


Engineering 
Library 


Entered  according  to  Act  of  Congress,  in  the  year  1892  by 

BROWN  &  SHARPE  MFG.  CO., 

In  the  Office  of  the  Librarian  of  Congress  at  Washington. 

Registered  at  Stationers'  Hall,  London,  Eng. 

All  rights  reserved. 


PREFACE. 


It  is  the  aim,  in  the  following  pages,  to  condense  as  much 
as  possible  the  solution  of  all  problems  in  gearing  which  in  the 
ordinary  practice  may  be  met  with,  to  the  exclusion  of  prob- 
lems dealing  with  transmission  of  power  and  strength  of 
gearing.  The  simplest  and  briefest  being  the  symbolical 
expression,  it  has,  whenever  available,  been  resorted  to.  The 
mathematics  employed  are  of  a  simple  kind,  and  will  present 
no  difficulty  to  anyone  familiar  with  ordinary  Algebra  and 
the  elements  of  Trigonometry. 


CONTENTS. 

FORMULAS    IN    GEARING. 


CHAPTER  I. 

PAGE 

Systems  of  Gearing .          .  .      . .       I 

CHAPTER    II. 

Spur  Gearing — Formulas — Table  of  Tooth   Parts — Comparative  Sizes 

of  Gear  Teeth  4 

CHAPTER  III. 

Bevel  Gears,  Axes  at  Right  Angles — Formulas— Bevel  Gears,  Axes  at 
any  Angle — Formulas — Undercut  in  Bevel  Gears — Diameter  Incre- 
ment— Tables  for  Angles  of  Edge  and  Angles  of  Face — Tables  of 
Natural  Lines n 

CHAPTER  IV. 
Worm  and  Worm  Wheel,  Formulas — Undercut  in  Worm  Wheels 32 

CHAPTER  V. 

Spiral  or  Screw  Gearing— Axes  Parallel — Axes  at  Right  Angles— Axes 

at  any  Angle — General  Formulas 36 

CHAPTER  VI. 
Internal  Gearing — Internal  Spur  Gearing — Internal  Bevel  Gears 41 

CHAPTER  VII. 
Gear  Patterns 47 

CHAPTER  VIII. 
Dimensions  and  Form  for  Bevel  Gear  Cutters 50 

CHAPTER  IX. 
Directions  for  cutting  Bevel  Gears  with  Rotary  Cutter 53 

CHAPTER    X. 
The  Indexing  of  any  Whole  or  Fractional  Number 56 

CHAPTER  XI. 

The  Gearing  of  Lathes  for  Screw  Cutting — Simple  Gearing — Compound 

Gearing — Cutting  a  Multiple  Screw   60 


ERRATA. 


P.  1 6.     Formula  "20=2  cos  a,"  should  read  2  a  =  2  s  cos  a. 

P.  57.     Example  III.     Instead  of  "we  advance  8  teeth  of  our 
147  tooth  gear,"  should  read — we  advance  87  teeth. 

P.  57.     Example  IV.       Instead    of    *'X=  '2    -f—  should 

190 

read   x=  12  4- —     This  would  make  it  necessary  to  ad- 

1 90 

vance  one  additional  tooth  at  a  time  of  the  change  gear  at 
60  even  intervals,  which  would  not  be  desirable;  but  if 
other  change  gears  were  on  hand,  say  with  88  or  95  teeth, 
better  results  would  be  obtained.  If  an  88  tooth  gear  were 
used  we  should  advance  one  turn  and  12  teeth  at  each 
indexing,  and  it  would  then  be  necessary  to  advance  an 
additional  tooth  at  only  8  intervals.  If  a  95  tooth  gear 
were  used  the  division  would  be  exactly  one  turn  and  13 
teeth  of  the  change  gear  with  no  correction  to  make. 

P.  62.     Fifth  paragraph.     Instead  of   ''gear  E   (being  fast  on 
same  shaft  with  E),"  should  read,  on  same  shaft  with  D. 

P.  62  &  63.     I.)  should  be  changed  to  E  in  all  formulas  under 
*•  Simple  Gearing." 

P.  65- 

Selecting      -I  G  =  30 


Change  "  E  =  50  "  to  E  ==  74. 

(OVEH). 


P.  66.  Second  paragraph  should  read, — Is  E  not  divisible  we 
find  how  many  turns  (V)  of  gear  R  are  made  to  each  full 
turn  of  the  spindle.  Dividing  this  number  by  2  for  double, 
by  3  for  triple  thread,  etc.,  we  advance  R  so  many  turns 
and  fractions  of  a  turn,  being  careful  to  leave  the  spindle 
at  rest. 

The  formula 

V  =  -    E 

for  simple  gearing,  might  be  omitted  as  there  would  be  no  case 
when  E  was  not  divisible  that  the  change  could  be  made  dt  R, 
and  it  is  considered  better  practice  to  make  the  change  at  E. 

The  rules  and  formulas  given  on  page  66  would  be  modi- 
fied when  the  gear  D  is  twice  as  large  as  the  gear  A  (as  explained 
in  the  fifth  paragraph  on  page  62)  and  to  provide  for  this  it 
would  be  necessary  to  divide  the  results  by  2  in  each  case. 

November,  1893- 


FORMULAS  IN  GEARING. 


BROWN  &  SHARPE  MFG.  CO. 

PROVIDENCE,  R.   I. 


FORMULAS   IN    GEARING. 


CHAF'TBR     I. 


SYSTEMS  OF  GEARING. 

(Figs,  i,  2.) 

There  are  in  common  use  two  systems  of  gearing,  viz.:  the 
involute  and  the  epicycloidal. 

///  1he  involute  system  the-outlines  of  the  working  parts  of  a 
tooth  are  single  curves,  which  may  be  traced  by  a  point  in  a 
flexible,  inextensible  cord  being  unwound  from  a  circular  disk 
the  circumference  of  which  is  called  the  base  circle^  the  disk 
being  concentric  with  the  pitch  circle  of  the  gear. 


In  Fig.  i  the  two  base  circles  are  represented  as  tangent  to 
the  line  P  P.  This  line  (P  P)  is  variously  called  "  the  line  of 
pressure,"  "  the  line  of  contact,"  or  "the  line  of  action." 


2  BROWN    &    SHARPE    MFG.    CO. 

In  our  practice  this  is  drawn  so  as  to  make  with  a  normal 
to  the  center  line  (O  O')  14^°,  or  with  the  center  line  75^°. 

The  rack  of  this  system  has  teeth  with  straight  sides,  the  two 
sides  of  a  tooth  making,  together,  an  angle  of  29°  (twice 


This  applies  to  gears  having  30  teeth  or  more.  For  gears 
having  less  than  30  teeth  special  rules  are  followed,  which  are 
explained  in  our  "  Practical  Treatise  on  Gearing." 


Fig.  2. 

In  epicycloidal,  or  double-curve  teeth,  the  formation  of  the 
curve  changes  at  the  pitch  circle.  The  outline  of  the  faces  of 
epicycloidal  teeth  may  be  traced  by  a  point  in  a  circle  rolling 
on  the  outside  of  pitch  circle  of  a  gear,  and  the  flanks  by  a  point 
in  a  circle  rolling  on  the  inside  of  the  pitch  circle.  The  faces 
of  one  gear  must  be  traced  by  the  same  circle  that  traces  the 
flanks  of  the  engaging  gear. 

In  our  practice  the  diameter  of  the  rolling  or  describing 
circle  is  equal  to  the  radius  of  a  i5-tooth  gear  of  the  pitch 
required  ;  this  is  the  base  of  the  system.  The  same  describing 
circle  being  used  for  all  gears  of  the  same  pitch. 


PROVIDENCE,    R.    I.  3 

The  teeth  of  the  rack  of  this  system  have  double  curves, 
which  may  be  traced  by  the  base  circle  rolling  alternately  on 
each  side  of  the  pitch  line. 

An  advantage  of  the  involute  over  the  epicycloidal  tooth  is, 
that  in  action  gears  having  involute  teeth  may  be  separated  a 
little  from  their  normal  positions  without  interfering  with  the 
angular  velocity,  which  is  not  possible  in  any  other  kind  of 
tooth. 

The  obliquity  of  action  is  sometimes  urged  as  an  objection 
to  involute  teeth,  but  a  full  consideration  of  the  subject  will 
show  that  the  importance  of  this  has  been  greatly  over-esti- 
mated. 

The  tooth  dimensions  for  both  the  involute  and  epicycloidal 
gears  may  be  calculated  from  the  formulas  in  Chapter  II. 


BROWN    &   SHARPE    MFG.    CO. 


II. 


SPUR    GEARING. 

(Figs.  3,  4.) 

Two  spur  gears  in  action  are  comparable  to  two  correspond- 
ing plain  rollers  whose  surfaces  are  in  contact,  these  surfaces 
representing  the  pitch  circles  of  the  gears. 

PITCH  OF  GEARS. 

For  convenience  of  expression  the  pitch  of  gears  may  be 
stated  as  follows  : 

Circular  pitch  is  the  distance  from  the  center  of  one  tooth  to 
the  center  of  the  next  tooth,  measured  on  the  pitch  line. 

Diametral  pitch  is  the  number  of  teeth  in  a  gear  per  inch  of 
pitch  diameter.  That  is,  a  gear  that  has,  say,  six  teeth  for  each 
inch  in  pitch  diameter  is  six  diametral  pitch,  or,  as  the  expres- 
sion is  universally  abbreviated,  it  is  "six  pitch."  This  is  by 
far  the  most  convenient  way  of  expressing  the  relation  of 
diameter  to  number  of  teeth. 

Chordal  pitch  is  a  term  but  little  employed.  It  is  the  dis- 
tance from  center  to  center  of  two  adjacent  teeth  measured  in 
a  straight  line. 


Fig.  3. 


PROVIDENCE,    R.    I. 

FORMULAS. 

N  =  number  of  teeth. 

s  =  addendum. 

/  =  thickness  of  tooth  on  pitch  line. 
/=  clearance  at  bottom  of  tooth. 
D"  =  working  depth  of  tooth. 
D"  +  /  =  whole  depth  of  tooth. 

d  =  pitch  diameter. 
d'  =  outside  diameter. 
P'  =  circular  pitch. 
Pc  =  chord  pitch. 
P  =  diametral  pitch. 
C  =  center  distance. 


p=^ 

P'  =  £F 


s=  —  =—  = 


c_.          - 

'   N        N  +  2 


2   P 


10 

s  4-  /  = 


20 
D"  =  25 

P.-,-.*: 

P'  =  d?r where  sin  d  =  — 

360°  •        d 


d'  =  d  +  2  s 


7t 


BROWN    &    SHARPE    MFG.    CO. 


GEAR  WHEELS. 

TABLE   OF    TOOTH   PARTS CIRCULAR    PITCH    IN    FIRST    COLUMN. 


Circular 
Pitch. 

Threads  or 
Teeth  per  inch 
Linear. 

Diametral 
Pitch. 

Thickness  of 
Tooth  on 
Pitch  Line. 

Addendum 
and  J^ 

1 
f 

o 

f1 

I 

ill 

PH 
O 

P' 
2 

*? 

'P 

t 

8 

D" 

•+/ 

D"+/ 

P'x.31  P'x.335 

i 

1.5708 

1.0000 

.6366 

1.2732 

.7366 

1.3732 

.6200 

.6700 

1J 

A 

1.6755 

.9375 

.5968 

1.1937 

.6906 

1.2874 

.5813 

.6281 

If 

4 

1.7952 

.8750 

.5570 

1.1141 

.6445 

1.2016 

.5425 

.5863 

!| 

A 

1.9333 

.8125 

.5173 

1.0345 

.5985 

1.1158 

.5038 

.5444 

1* 

1 

2.0944 

.7500 

.4775 

.9549 

.5525 

1.0299 

.4650 

.5025 

!iV 

41 

2.1855 

.7187 

.4576 

.9151 

.5294 

.9870 

.4456 

.4816 

If 

A 

2.2848 

.6875 

.4377 

.8754 

.5064 

.9441 

.4262 

.4606 

!iV 

if 

2.3936 

.6562 

.4178 

.8356 

.4834 

.9012 

.4069 

.4397 

1* 

4 

2.5133 

.6250 

.3979 

.7958 

.4604 

.8583 

.3875 

.4188 

ifV 

•H 

2.6456 

.5937 

.3780 

.7560 

.4374 

.8156 

.3681 

.3978 

4 

f 

2.7925 

.5625 

.3581 

.7162 

.4143 

.7724 

.3488 

.3769 

XA 

W 

2.9568 

.5312 

.3382 

.6764 

.3913 

.7295 

.3294 

.3559 

1 

i 

3.1416 

.5000 

.3183 

.6366 

.3683 

.6866 

.3100 

.3350 

H 

IA 

3.3510 

.4687 

.2984 

.5968 

.3453 

.6437 

.2906 

.3141 

i 

H 

3.5904 

.4375 

.2785 

.5570 

.3223 

.6007 

.2713 

.2931 

if 

XA 

3.8666 

.4062 

.2586 

.5173 

.2993 

.5579 

.2519 

.2722 

f 

ii 

4.1888 

.3750 

.2387 

.4775 

.2762 

.5150 

.2325 

.2513 

•H 

IJT 

4.5696 

.3437 

.2189 

.4377 

.2532 

.4720 

.2131 

.2303 

.1 

14 

4.7124 

.3333 

.2122 

.4244 

.2455 

.4577 

.2066 

.2233 

PROVIDENCE,    R.    I. 


TABLE  OF  TOOTH  PAKTS.— Continued. 

CIRCULAR   PITCH    IN    FIRST    COLUMN. 


eS 

1/2  *•<  S 

rrt  OJ  2^ 

1  . 

en  — 

t-i 

0  ^  « 
g  5.S 

1 

-0  -t 

l| 

ii 

Ii 

°I-d 

*t 

0  .2 

l&J 

sS 

||^ 

a 

O  "^ 

bfio 

"o^^ 

W  o 

C  LL 

6^|  2J3 

5^ 

£3^ 

q 

Is 

«|2 
|HS 

T3  a 

5s 

r 

ll 

l? 

h 

P 

P' 

»r 

P 

« 

i 

D" 

•+/ 

D"+/ 

P'x.31  P'x.335 

| 

If 

5.0265 

.3125 

.1989 

.3979 

.2301 

.4291 

.1938 

.2094 

A 

M 

5.5851 

.2812 

.1790 

.3581 

.2071 

.3862 

.1744 

.1884 

4 

2 

6.2832 

.2500 

.1592 

.3183 

.1842 

.3433 

.1550 

.1675 

TV 

2f 

7.1808 

.2187 

.1393 

.2785 

.1611 

.3003 

.1356 

.1466 

! 

91 
2 

7.8540 

.2000 

.1273 

.2546 

.1473 

.2746 

.1240 

.1340 

I 

2f 

8.3776 

.1875 

.1194 

.2387 

.1381 

.2575 

.1163 

.1256 

4 

3 

9.4248 

.1666 

.1061 

.2122 

.1228 

.2289 

.1033 

.1117 

iV 

3* 

10.0531 

.1562 

.0995 

.1989 

.1151 

.2146 

.0969 

.1047 

f 

8i 

10.9956 

.1429 

.0909 

.1819 

.1052 

.1962 

.0886 

.0957 

i 

4 

12.5664 

.1250 

.0796 

.1591 

.0921 

.1716 

.0775  .0838 

I 

44 

14.1372 

.1111 

.0707 

.1415 

.0818 

.1526 

.0689  .0744 

j 

5 

15.7080 

.1000 

.0637 

.1273 

.0737 

.1373 

.0620  .0670 

iV 

5J 

16.7552 

.0937 

.0597 

.1194 

.0690 

.1287 

.0581 

.0628 

{ 

6 

18.8496 

.0833 

.0531 

.1061 

.0614 

.1144 

.0517 

.0558 

1 

7 

21.9911 

.0714 

.0455 

.0910 

.0526 

.0981 

.0443 

.0479 

} 

8 

25.1327 

.0625 

.0398 

.0796 

.0460 

.0858 

.0388 

.0419 

I 

9 

28.2743 

.0555 

.0354 

.0707 

.0409 

.0763 

.0344 

.0372 

T1o 

10 

31.4159 

.0500 

.0318 

.0637 

.0368 

.0687 

.0310 

.0335 

A 

16 

50.2655 

0312 

.0199 

.0398 

.0230  .0429 

.0194 

.0209 

BROWN    &    SHARPE    MFG.    CO. 

GEAR  WHEELS. 

TABLE  OF  TOOTH  PARTS DIAMETRAL    PITCH   IN    FIRST    COLUMN. 


Diametral 
Pitch. 

Circular 
Pitch. 

Thickness 
of  Tooth  on 
Pitch  Line. 

Addendum 

and  ^-' 

i 

f1 

&5 
«J 

P 

P' 

t 

s 

D" 

*+/. 

D"+f. 

i 

6.2832 

3.1416 

2.0000 

4.0000 

2.3142 

4.3142 

f 

4.1888 

2.0944 

1.3333 

2.6666 

1.5428 

2.8761 

1 

3.1416 

1.5708 

1.0000 

2.0000 

1.1571 

2.1571 

1J 

2.5133 

1.2566 

.8000 

1.6000 

.9257 

1.7257 

1J 

2.0944 

1.0472 

.6666 

1.3333 

.7714 

1.4381 

If 

1.7952 

.8976 

.5714 

1.1429 

.6612 

1.2326 

2 

1.5708 

.7854 

.5000 

1.0000 

.5785 

1.0785 

01 

1.3963 

.6981 

.4444 

.8888 

.5143 

.9587 

o  i 

1.2566 

.6283 

.4000 

.8000 

.4628 

.8628 

2J 

1.1424 

.5712 

.3636 

.7273 

.4208 

.7844 

3 

1.0472 

.5236 

.3333 

.6666 

.3857 

.7190 

8J 

.8976 

.4488 

.2857 

.5714 

.3306 

.6163 

4 

.  7854  . 

.3927 

.2500 

.5000 

.2893 

.5393 

5 

.6283 

.3142 

.2000 

.4000 

.2314 

.4314 

C 

.5236 

.2618 

.1666 

.3333 

.1928 

.3595 

7 

.4488 

.2244 

.1429 

.2857 

.1653 

.3081 

8 

.3927 

.1963 

.1250 

.2500 

.1446 

.2696 

9 

.3491 

.1745 

.1111 

.2222 

.1286 

.2397 

10 

.3142 

.1571 

.1000 

.2000 

.1157 

.2157 

11 

.2856 

.1428 

.0909 

.1818 

.1052 

.1961 

12 

.2618 

.1309 

0833 

.1666 

.0964 

.1798 

13 

.2417 

.1208 

.0769 

.1538 

.0890 

,.1659 

14 

.2244 

.1122 

.0714 

.1429 

.0826 

.1541 

PROVIDENCE,  R.    I. 

TABLE  OF  TOOTH  PARTS—  Continued. 

DIAMETRAL   PITCH    IN   FIRST    COLUMN. 


Diametral 
Pitch. 

Circular 
Pitch. 

Thickness 
of  Tooth  on 
Pitch  Line. 

Addendum 
and  ^- 

f 

&i 
si 

V* 

* 

P«  « 
02  fe.S 

•M  §  Hi 

:*i 

is 

h 

Po 

^E-. 

1' 

P. 

P'. 

t. 

s. 

D". 

*+/. 

D"4-/. 

15 

.2094 

.1047 

.0666 

.1333 

.0771 

.1438 

16 

.1963 

.0982 

.0625 

.1250 

.0723 

.1348 

17 

.1848 

.0924 

.05^8 

.1176 

.0681 

.1269 

18 

.1745 

.0873 

.0555 

.1111 

.0643 

.1198 

19 

.1653 

.0827 

.0526 

.1053 

.0609 

.1135 

20 

.1571 

.0785 

.0500 

.1000 

.0579 

.1079 

22 

.1428 

.0714 

.0455 

.0909 

.0526 

.0980 

24 

.1309 

.0654 

.0417 

.0833 

.0482 

.0898 

26 

.1208 

.0604 

.0385 

.0769 

.0445 

.0829 

28 

.1122 

.0561 

.0357 

.0714 

.0413 

.0770 

30 

.1047 

.0524 

.0333 

.0666 

.0386 

.0719 

32 

.0982 

.0491 

.0312 

.0625 

.0362 

.0674 

34 

.0924 

.0462 

.0294 

.0588 

.0340 

.0634 

36 

.0873 

.0436 

.0278 

.0555 

.0321 

.0599 

38 

.0827 

.0413 

.0263 

.0526 

.0304 

.0568 

40 

.0785 

.0393 

.0250 

.0500 

.0289 

.0539 

42 

.0748 

.0374 

.0238 

.0476 

.0275 

.0514 

44 

.0714 

.0357 

.C227 

.0455 

.0263 

.0490 

46 

.0683 

.0341 

.0217 

.0435 

.C252 

.0469 

48 

.0654 

.0327 

.0208 

.0417 

.0241 

.0449 

50 

.0628 

.0314 

.0200 

.0400 

.0231 

.0431 

56 

.0561 

.0280 

.0178 

.0357 

.0207 

.0385 

60 

.0524 

.0262 

.0166 

.0333 

.0193 

.0360 

OF. 


'TTHIVERSITY; 


10 


BROWN    &    SHARPS    MFG.    CO. 


Comparative  Sizes  of  Gear  Teeth, 
Involute. 


8  p 


Fig.  4. 


9  P 


PROVIDENCE,    R.    ]. 


I  I 


CHAF>TKR   in. 
BEVEL  GEARS.— AXES  AT   RIGHT  ANGLES. 

(Fig.  5«) 


12  BROWN    &    SHARPE   MFG.    CO. 


FORMULAS. 


N»=   |  Number  of  teeth  ) 

P  =  diametral  pitch. 
P'  =  circular  pitch. 

aa  =   \  center  angle  =  angle  of  edge  j  gear. 
ab  =   \  or  pitch  angle  (  pinion. 

ft  =  angle  of  top. 
ft'  —  angle  of  bottom. 

g=  [angle  of  face  {  ^on 


A  =  apex  distance  from  pitch  circle. 
A'  =  apex  distance  from  large  bottom  of  tooth. 
d  =  pitch  diameter. 
d'  =  outside  diameter. 
s  =  addendum. 

/  =  thickness  of  tooth  at  pitch  line. 
f  =  clearance  at  bottom  of  tooth. 
D"  =  working  depth  of  tooth. 
+/=  whole  depth  of  tooth. 
2  a  =  diameter  increment. 

b  =  distance  from  top  of  tooth  to  plane  of  pitch  circle. 
F  =  width  of  face. 


PROVIDENCE,    R.    I. 


13 


tan  ft  =  JL 


N 


N 


ton       = 


£•„  =  90°  -  (aa  +  /?)  ;  gb  =  90°  -  (flf6  +  ft) 
h  —  a  —  ft'         (See  Note,  page  5  2.) 


A  — 

N 

\2 

A'-               N 

A' 

2  P  sin  a 
A 

A  ^^ 

p  = 

cos  ^' 
*            cos  / 

2  P  sin  a  cos  // 
? 

sin  (a  +  p) 

N 

2  A  sin  or 

A 


2  ^  =  2  j  cos  a' 


=  a  tan 


(See  page  20.) 

a  for  gear      =  b  for  pinion 
^  for  ^inion  =  ^  for  gear 


P' 


=  .3183?'     J  =  Atan/? 
'  j+/=Atan/J' 


IP 


/-   W 


UITI7BRSIT7' 


NOTE.—  Formulas  containing  notations  without  the  designating  letters  a  and  b 
apply  equally  to  either  gear  or  pinion.  If  wanted  for  one  or  the  other,  the  respective 
letters  are  simply  attached. 


BROWN    &    SHARPE    MFG.    CO. 


BEVEL  GEARS  WITH  AXES  AT  ANY  ANGLE. 

(Figs.  6,  7.) 


Pinion 


f      Fiff.  0. 


PROVIDENCE,    R.    I.  15 


FORMULAS. 

C  =  angle  formed  by  axes  of  gears. 

J"Ja  =   i  number  of   teeth  -I  g?a.r' 
Nft  —    \  I  pinion. 

P  =  diametral  pitch. 
P'  =  circular  pitch. 

%  =   \  angle  of  edge  =  pitch  angle  {  ^ 

ft  —  angle  of  top. 
f$  =  angle  of  bottom. 


£=}  angle  of  face  { 


£  =   |  cutting  angle 

A  =  apex  distance  from  pitch  circle. 
A'  =  apex  distance  from  large  bottom  of  tooth. 

d  ==  pitch  diameter. 
d'  =  outside  diameter. 
2  a  =  diameter  increment. 

b  =  distance  from  top  of  tooth  to  plane  of  pitch  circle. 


NOTE.  —  The  formulas  for  tooth  parts  as  given  on  page  5  apply  equally  to  these 
cases. 


l6  BROWN    &    SHARPE    MFG.    CO. 


smC 

N 


tanga==  ;orcot<ra  =  --»—  -H  cot  C 

*»  N°smC 


tan  <xb  =    .   SmC      ;  or  cot  ab  =       N«       4  cot  C 


NOTE.—  These  formulas  are  correct  only  for  values  of  C  less  than  90°.     If  C  is 
greater  than  90°,  consult  the  following:  page. 


„       2  sin  a  0        5 

tan  ft  =  —  -  —  ;     or  tan  ft  =  —  ; 


ga  =  90°  -  K+  /?)  ;  ^6  =  90°  -  (ab 


A=        N 


2  P  sin  a 


A'=      A 


COS  ft' 


P  7T 

2  tf  =  2SCOS  « 

#  for  gear      —  /^  for  pinion. 
a  for  pinion  =  b  for  gear. 

NOTE.  —  See  Foot  Note  on  page  13. 


PROVIDENCE,    R.    I. 


l8  BROWN    &   SHARPE    MFG.    CO. 


The  formulas  given  for  aa  and  ctb  (when  C,  Na  and  N6  are 
known)  undergo  some  modifications  for  values  of  C  greater 
than  90°. 

For  bevel  gears  at  any  angle  but  90°  we  may  distinguish 
four  cases  ;  C,  Na,  N6  being  given. 

/.  Case.     See  pages  14  and  16. 

//.  Case.     C  is  greater  than  90°. 

tan  ora=  — —  — L  ;     tan  ab  =  _ — 

_6-cos(i8o-C)  S-a~cc 

Na  N6 

///.  Case.     aa  —  90°  ;  ab  =  C  —  90° 

IV.  Case. 

sin  E  sin  E 


cos  E  -  ^  £?  -  cos  E 

Na  N6 

For  an  example  to  apply  to  Case  III.,  the  following  condi- 
tion must  be  fulfilled  : 

Na  sin  (C  -  90°)  =  Nb 

To  distinguish  whether  a  given  example  belongs  to  Case  II. 
or  case  IV.,  we  are  guided  by  the  following  condition  : 

T      XT     .     ,~  0\  (  smaller  than  N6,  we  have  Case  II. 

Is  :  Na  sin  (C  -  9°  )  \  larger  than  NJ'We  have  case  IV. 


PROVIDENCE,    R.    I.  -       19 


UNDERCUT  IN  BEVEL  GEARS. 

By  undercut  in  gears  is  understood  a  special  formation  of 
the  tooth,  which  may.be  explained  by  saying  that  the  elements 
of  the  tooth  below  the  pitch  line  are  nearer  the  center  line  of 
the  tooth  than  those  on  the  pitch  line.  Such  a  tooth  outline  is 
to  be  found  only  in  gears  with  few  teeth.  In  a  pair  of  bevel 
gears  where  the  pinion  is  low-numbered  and  the  ratio  high,  we 
are  apt  to  have  undercut.  For  a  pair  of  running  gears  this 
condition  presents  no  objection.  Should,  however,  these  gears 
be  intended  as  patterns  to  cast  from,  they  would  be  found  use- 
less, from  the  fact  that  they  would  not  draw  out  of  the  sand. 
We  have  stated  on  page  2  (see  Fig.  i)  that  the  base  of  our 
involute  system  is  the  14^2°  pressure  angle.  If  a  pair  of  bevel 
gears  with  teeth  constructed  on  this  basis  have  undercut,  we 
can  nearly  eliminate  the  undercut  —  and  for  the  practical  work- 
ing this  is  quite  sufficient—  by  taking  as  a  basis  for  the  con- 
struction of  the  tooth  outline  a  pressure  angle  of  20°. 

The  question  now  is  :  When  do  we,  and  when  do  we  not 
have  undercut  ?  Let  there  be  : 

N  =  number  of  teeth  in  gear. 
n  =  number  of  teeth  in  pinion. 

n  V  N9  +  »a  ^_  * 


N 
where  we  have  undercut  for/  less  than  30. 

This  formula  is  strictly  correct  for  epicycloidal  gears  only. 
It  is,  however,  used  as  a  safe  and  efficient  approximation  for 
the  involute  system. 


20 


BROWN    &    SHARPE    MFG.    CO. 


DIAMETER    INCREMENT. 

2  a. 

RULE. — The  ratio  being  given  or  determined,  to  find  the  outside  diameter 
divide  figures  given  in  table  for  large  and  small  gear  by  pitch  (P)  and  add 
quotient  to  pitch  diameter. 


RATIO. 

GEARS. 

RATIO. 

GEARS. 

RATIO. 

GEARS. 

Large 

Small 

Large 

Small 

Large 

Small 

1.00 

1:1 

1.41 

1.41 

1   65 

1.05 

1.70 

4.40 

.45 

1.94 

1.05 

1.37 

1.42 

1.67 

5:3 

1.03 

1.72 

4.50 

9:2 

.44 

1.95 

.07 

1.36 

1.43 

1.70 

1.01 

1.73 

4.60 

.42 

1  95 

.10 

1.35 

1.44 

1.75 

7:4 

.99 

1.74 

4.80 

.41 

1.96 

.11 

10:9 

1.34 

1.46 

1.80 

9:5 

.97 

1.75 

5  00 

5:1 

.39 

1.96 

.12 

1.33 

1  46 

1.85 

.95 

1.76 

5.20 

.38 

1.96 

.13 

9:8 

1  33 

1.47 

1.90 

.93 

1  77 

5.40 

.37 

1.96 

.14 

8:7 

1.32 

1.49 

1  95 

.91 

1.78 

5.60 

.36 

1.97 

.15 

1.31 

1.50 

2  00 

2:1      .89 

1.79 

5.80 

.34 

1.97 

.16 

1.30 

.51 

2  10 

.87 

1.80 

6  00 

6:1 

.33 

1.97 

.17 

7:6 

1.30 

.52 

2.20 

.84 

.81 

6.20 

.32 

1  97 

.18 

1.29 

.53 

2  25 

9:4 

.82 

.82 

6  40 

.31 

1.97 

.19 

1.28 

53 

2.30 

.80 

.83 

6.60 

.30 

1  97 

.20 

6:5 

1.28 

.54 

2.33 

7:3 

.78 

.84 

6.80 

.29 

1  98 

.23 

1.27 

.55 

2.40 

.76 

.85 

7  00 

7:1 

.28 

1.98 

1.25 

5:4 

1.25 

.56 

2.50 

5:2 

.75 

.86 

7.20 

.27 

1.98 

1.27 

1.25 

.57 

2.60 

.73 

.86 

7.40 

.27 

98 

1.29 

9:7 

1.24 

.58 

2.67 

8:3 

.71 

1.87 

7.60 

.26 

98 

1.30 

1.22 

.59 

2.70 

.69 

.87 

7  80 

.26 

.98 

1.88 

4:3 

1.20 

.60 

2.80 

.67 

.88 

8  00 

8:1 

.25 

98 

1.35 

1.18 

.61 

2.90 

.65 

1.89 

8.20 

.24 

.98 

1  37 

1.17 

1.61 

3.00 

3:1 

.63 

1.91 

8  40 

.24 

.98 

1.40 

7:5 

1.16 

1.62 

3.20 

.60 

1.92 

8.60 

.23 

1.98 

1.43 

10:7 

1.15 

1.63 

3.33 

.58 

1.92 

8.80 

.23 

1  98 

1.45 

1.13 

1.65 

3.40 

.56 

1  92 

9.00 

9:1 

.22 

1.99 

1.50 

3:2   1.11 

1.66 

3.50 

7:2 

.54 

1.93 

9.20 

.22 

1.99 

1.53 

1.10 

1.67 

3.60 

.52 

1  93 

9.40 

.21 

1.99 

1  55 

1.09 

1.67 

3.80 

.50 

1.94 

9.60 

.21 

2.00 

1.58 

1.08 

1.68 

4.00 

4:1 

.49 

1.94 

9.80 

.20 

2.00 

1.60 

8:5 

1.07 

1.68 

4.20 

.47 

1.94 

10.00 

10:1 

.20 

2  00 

NOTE. — To  be  used  only  for  bevel  gears  with  axes  at  right  angle. 


PROVIDENCE,    R.    I.  21 


TABLES  FOR  ANGLES  OF  EDGE  AND  ANGLES 

OF  FACE. 

The  following  three  tables  have  been  computed  for  the 
convenience  in  calculating  datas  for  bevel  gears  with  axes  at 
right  angle.  They  do  not  hold  good  for  bevel  gears  with  axes 
at  any  other  angle. 

To  use  the  tables  the  number  of  teeth  in  gear  and  pinion 
must  be  known. 

Having  located  the  number  of  teeth  in  the  gear  on  the 
horizontal  line  of  figures  at  the  top  of  the  table,  and  the  num- 
ber of  teeth  in  the  pinion  on  the  vertical  line  of  figures  on  the 
left-hand  side,  we  follow  the  two  columns  to  the  square  formed 
by  their  intersections. 

The  two  angles  found  in  the  same  square  are  the  respective 
angles  for  gear  and  pinion.  The  tables  are  so  arranged  that 
the  angle  belonging  to  the  gear  is  always  placed  above  the 
angle  for  the  pinion. 


22 


BROWN    &    SHARPE   MFG.    CO. 


TABLE  i 
ANGLE  OF  EDGE. 


41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

12 

73°4i 

6*19 

73\8 

6*46 

72*54 

I7Y 

7Z*aa 

17*32 

IZ'z 

17*58 

71  3* 

18*86 

71*5' 
18*55 

7034 
19*26 

7o'i 

13*59 

69"26 

20V 

63» 
21*o' 

68'ie 

2  1  48 

67V 
22'e? 

66*48 

23*12 

66V 

23*58 

13 

78  tS 

17*35 

/i  r 

l«*t 

71  34 

18*26 

71*7 
18*53' 

7039 

I9*ei' 

70*3 
19V 

69*37 

20*w 

69's 

80*55 

6B»o 

21*30 

67  S3 
22*7' 

67'ib 

22*45 

66*34 
23*26 

65V 

24*9 

65'6 

24*54 

64*17' 

25*43 

14 

71    9 

18  V 

7043 
19*17 

70  is 

13*45 

69*46 

80*14 

63  is 

20*44 

6845 

2  1  *is 

68*te 

21*48 

6737 
22« 

67"o 
23*0 

66  2i 

23*37 

6548 

e4*m' 

64*59 

25*  r 

64"(4' 
25*4* 

63  "26 

26*34 

62*36 
27*24 

15 

69'54 
20*6' 

6y'z 
20*3*. 

68*5$ 

21  V 

68f8 

2IS21 

6/56 

22*4 

67*23' 

22V 

66*44 

23*8 

66*12 
234« 

65  si 

24*27 

64  S3 
£5*7' 

64% 
25*50 

63*24 
26*34' 

62*J9 
27*21 

61*48 

28"r 

60's* 
29B 

16 

68V 

21*19 

68^1 

?l  48 

67V 

eaV 

67*0 
22*50 

6637 
23*o 

&6z 
23*58 

feS'ee 

24*34 

6448 
25*12 

64"  8 

25*58 

63» 
26V 

6242 

27*» 

61*%' 
28*4' 

61  S 
28*5i 

60°I5 

29*45 

59*2i' 

30*33 

17 

6729 

28*3  i' 

66  sa 
23*e 

66°27 

E3*31 

65*54 
2<V*6 

65  19 
84*4t' 

64*43' 
25*7 

64'  6 

25*54 

6326 
26*34' 

6245 

27*is 

ee'r 

27*59 

61*5 
28*45 

feo'w 

29*32 

59*37 

30*n 

584* 

3J*6 

5748 

32*12 

18 

66*18 
23*4. 

654 

24*14 

65*14 

24*44 

64's9 
25*8. 

64J4' 
25V 

63*H 

26*54 

62°47 
27*ii 

62V 

27*54 

ei'o' 

28*s?' 

60>>3« 
29*Ei 

59V 
30*9 

59*2 

30*s« 

58*io 

SI*M 

57"|6 

32*44 

S6V 
33*4i 

19 

65V 

64"j«, 

64V 

63*efc 

6e°<» 

62*io 

6i'» 

60°48 

60*4 

S9Ji8 

58"sd 

57°39' 

564« 

5&V 

54*52 

35V 

24  sa 

25  «4 

esss 

26  sw 

27  i. 

27  so 

28  so 

29* 

29  56 

3042 

31  30 

32  ei 

33(4 

349 

20 

164V 
26V 

€3** 

26*34 

ee'si 

27°9 

62'i4 

e7**fe 

61  "37 
28°23' 

60°S7 
29** 

60uis 
29*45 

SS'az' 

30*28 

5847' 
31*13 

58"  o' 
32°o' 

57UK> 
32*50 

56°  19 
33*41 

55"24 

34*% 

5V28 
35*32 

53*28 
36*3?' 

21 

62'si 
27Y 

62'» 
27*4t 

61*42 

28*rt 

61*4' 

28°w 

60J25 

29*js 

5344S 
30*.s' 

59'  i 

30*a 

58*18 
31*42 

57  '32 

32*Z6 

5643 

33*17 

SS^U 
34*7' 

55Jo 
35*o' 

54<>5 
35*ss 

53*7 

36*53 

sa's 
37a 

22 

SlV 

61°  I 

60"34 

59S6 

59*15 

58»4 

57"si' 

576' 

56'is 

55U23 

54*58 

53*45 

52*43 

51*50 

5049 

28  .3 

284_ 

2926 

304 

3045 

31  26 

329 

32  S4 

334i 

3431 

35  22 

36i5 

37n 

38io 

39)1 

23 

60*4i 

60*6 

59*88 

b«4S 

58U8 

57*25 

54"  16 

53*24 

52*3i' 

51*15 

5036 

43"»» 
40e2«; 

t*J>8 

29  5* 

3032 

31  ii 

31  52 

32  as 

33» 

34s 

3453 

3541 

36  M. 

3729 

38  25 

3924 

24 

153*39 

53*a 

58*« 

57*44 

57U  z 

56*19 

55*33 

54°47' 

53*S8 

53*7' 

52*15 

51*20' 

50*23 

4324 

48*K 

30  zi 

30  s» 

31  37 

32  i« 

32  w 

334i 

34  27 

3b<3 

36z 

36  S3 

3745 

3840 

3937 

40*3* 

4IM 

25 

58» 

58"  o 

57  eo 

56"4o 

55J57 

S5°i3 

54*Z8 

53J4o 

52V 

52V 

5lV 

50°i2 

49V 

48*14 

47Jt2 

31  22 

320 

3240 

33w 

J43 

M47 

3532 

3620 

37  9 

38o 

3853 

3346 

40  46 

4146 

4<i48 

26 

5737' 

56s 

5€'n 

55*37 

54*S4 

54°  id 

53*24 

5236 

51*46 

5054 

sojr 

43*  & 

46U7 

47% 

46*5 

3223 

33  e 

334i 

3423 

35  6 

35so 

36  3* 

3/Jv 

38)4 

396 

39  S3 

4055 

41  53 

4^53 

4355 

27 

56*38 

55°S9 

ss1* 

54°36 

53"S3 

53U7 

52JZI 

sT^j 

5043 

43USI 

48US7 

4€uo' 

47*3 

46J2 

4-5* 

33  22 

34 

3442 

3524 

367 

3bs3 

3/39 

3«e7 

3317 

409 

41  3 

42  o 

42  57 

43  58 

28 

S5*4o 

55* 

54*19 

53*37 

52*53 

52<is 

5I°20 

50*32 

49c«i 

48*48 

47*55 

46*58 

46*0 

45* 

3420 

35o 

3b4l 

3*m 

3/7 

3/sz 

3840 

33M 

40  is 

41  12 

42  5 

43a 

440 

29 

54V 
35*16 

54*3 
35*s 

53*a 

36*38' 

5EL39 

37*er 

51*55 
38*  S 

51*9 
38°si 

50"2i' 
39*39 

43^32 
40*28 

48"4i 
4I*» 

47^' 
42*  n' 

4€US4 
43*6 

45  %« 
44*2' 

45* 

30 

S3*4« 
36'tz 

53*7 

36*53 

S2*ts 

37°M 

51*42 

38*8 

50°S8 

39V 

SO'ie 

49*24 

48L3S 

4743 

46"si 

45*56 

45° 

3346 

40  M 

4125 

42  17 

439 

444 

31 

5254 

376 

58* 

5l"3t 

50*48 

50"» 

43u|fe 

48°2S 

47M 

4647 

45's-V 

45* 

374 

3*  a 

39  it 

39  se 

4044 

4-1  32 

42  ti 

43)3 

446 

32 

52*2 

Si'a 

50°M 

49V 

49*9 

48*n 

47V 

46*44 

45*53 

45° 

37s* 

3tt4 

3922 

406 

40  st 

4I*» 

42  a, 

43*16 

447 

33 

Sl'io 

50*2, 

49U46 

43'z 

48*ifc 

47*» 

4641 

45s. 

45° 

38  so 

333 

40(4 

40  s» 

41  44 

42  ai 

43°i» 

44  9 

34 

50*20 

493 

48"S5 

48°)i 

47*2S 

46's« 

4Slso 

45* 

39*40 

40  K 

41   5 

41*49 

42  35 

43*22 

44(0 

35 

49*3 

48V 

46*S 

47*d 

46U35 

45*48 

45° 

4 

4023 

41  i 

41  ss 

4239 

43  tS 

44)2 

36 

4843 

41*17 

48° 
42* 

47*7 

4e*43 

46*33 
43*27 

4547 

44*« 

45° 

37 

47°56 

42*4 

4?" 
42*4< 

46°30 
43*30 

454* 
44V 

45° 

38 

47*io 
42*50 

4fe' 
43M 

45°4S 
44*is 

45* 

39 

46** 
43M 

45v 

44° 

45* 

40 

45*42 

44*,8 

45 

4 

45* 

PROVIDENCE,    R.    I. 


TABLE  i.— (Continued.) 


ANGLE  OF  EDGE. 


page 


24 


BROWN    &    SHARPE    MFG.    CO. 

TABLE    2. 
ANGLE  OF  EDGE. 


PROVIDENCE,    R.    I. 

TABLE   2.— (Continued) 
ANGLE  OF  EDGE. 


565554535251 


5049484746454443 


42 


1? 
13 
14 

15 
16 
17 

18 
19 


77  54  7742 


12  « 


7728 

12*32 


77  is  77  o 


76*46 


76V  7558  7541 


13   14 


75*23 
14*37 


7445  74  as 

16*35 


743 
15*57 


13*4 


7642 

13*18 


1332 


1347 


7S°4C 
14*18 


t4S2 


74  si 
1S°9 


7353  73» 


»b7 


73 

16*49 


17  re 
71*34 


75*59 

14*4 


75*43 
M*I7 


75*28 
I4*3i 


75K 

14*48 


7421  74°3 
IS°39|  fS*B7 


16*35 


73*4' 

16*5* 


17*17' 


17*39 


reV 


75V 


7444 
15*16 


74°» 
15*31 


7**S9 

1642117' 


723972(8 


71*56 
t8*4 


71*3* 


71*  K> 


70*4i 


70V 
19» 
69*9 

20*51 


1557 


7347 

16*13 


73 
16*36 


73°ii 

16*48 


72*54J  72*35  72 
17*25 


71  3* 

I8°26  I8"48 


702670° 

19*34  19*59'!  20*a» 


7249 

•7V 


72*13 
17*47 


71*54  7I°3 
18*6 


3*1 7 1*13  70*52 
18*47    19*8 


70*30  70*7 


694369' 


19  30  19  53  2O  17 


20*452 


22V 
66*48 


72 
17  49 


71*53 
18*7 


71: 

18*26 


71*15 
18*45 


70*s» 


70"33 


69'so 


69°Z6  69  3 


67*17 


7l*.s 


7057 

is*  r 


70*37 
19*23 


70  17  69 


69 


1943 


204- 


20  z« 


68  4« 
2'"« 


685 


21  35 


22  i 


6734 
22*26 


67*6 
22*54 


66  »8  fee  10 


23 


23so 


65*39' 
2»"2 
64*32' 


70°ZI  70°  l 


69*4  1' 


69  19  66*57  68  35  68*2  67 4»  67  23  6657  66*30  66* a 


22 


23*3 


65"  33)  65*  3 

a**n 


2 


mwTV 


2O*3*  20V  21* 


68*45  68*23  68°o 


22*47  23*12' 


6fi*4e  66«  fcS°ss  65*28  64*59 


2338 


24S 


25  i' 


25*3 


26  a 


63*26 
26*3* 

6?27 
27*39 


68  S3  68  ia  67*50  67*27  67e4 


66*40 


21  272148 


22  33  22  56  23  20 


63*49 

23*45|  24*  n1 


65  M  64  55 


64*16  63*57  63 


24  J7  25  S 


25*3* 


26* 


263* 


62*54 
27*6 


23 

24 
25 
26 


67*41 
22*19 


66 32  66*8 


23  5 


23  28  23  52 


65*44 

24*16 


65*18  64*51 
24-J25V 


64*24  63  55 


25*36  26*s 


62*56  62*24  61*52 


27 


27s* 


66*48  66  86  66*  z 
23°K>  23*34  23*58 


65*38  6S°|4  64*48|64*tt|63s* 
24*« 


25*3 


63*26  62*57  62* 


61-56  61*23 


60*5^ 


284 


28  S7 


60*15 
2945 


65  57  65  n 


243 


2*27 


65*9 

24V 


64*45  64* 
ZS'.s  25*40  2fe' 


63*5 

v 


26*34^  27 V 


62*58  62*29  61*59  61*29  60*57  60V 


2731 


28 


28*3 


29*3 


29  s«  30io 


30*46 

SB**: 

31*46 


65*6 


6442  64*18  63*5*  63  26  62*59  62*3. 
26*3427*1 


25'w25*- 


268 


62*3 

27"2«|  27*57 


59*59  59*25  58  M 


28t7  285729W30  t 


30S5 


63'si 


63t6j63o 
27Y 


27*2* 


62*6 


60  M  ttri 

»  29*53 


59*35 


59  1 


58  a  57*53 

32  32*7 


3t*» 
S?» 


26*34  26*S»  27*24  27*si' 


61* 

28*i»'|26*46 


14  60*45  60*15  594S59  13  SCT49  58*7 


tfdtfm 


57° 


30*.5  3047  3!°2«'  31*53  32' 


56*56 
33 


6Ti2  61  45  6r  19  6051 
29V 


27*23  27*4*  28' 


29*37  30*7  30*37 


59  23  58 '52  Sr»  5r*6  57  it  56*37  5*T* 
31*8  31*41  32*14  32%i  33*t3  34* o 


SS*a 

34*»7 


6I**»  ftTtt  60  S, 


5>>aS9z  SB*  58  0 
30*28  30*S8j3r2»  32*  o' 


5r*J  56°53  56*19  55Vi  5S'» 


33*7 


33*4,'  34' 


oi*** 

31  Lew 


6016  60  c 


28*»  MV  t9' 


59iaS84258i257V 


594» 
s*|  30*19  30-48131 


31*4 


57*  •' 


•CM  set 


SS*«  54  s«  54*  it 


32  «  33  M  33  59  3434  35  |0 


35*48 


53*»4 


59*46  5911 
29"45  30*ii  30*39  31*8 


*26  32*6 


56*51  56  19  5545  55 


33*8 


33**i  34*i5  34*49  35*u  36* a 


5929592 


58»  58's 


57  36  57  6  56' 


54*5654*21 


B345  53  8 


SZ*t9 


30's»  31*2*  SI'SS  32  v> 


33583430354. 


35*39  36  is 


36*52  3f7*3i' 


5I°50 

38V 


34  iS 


58V  58  is  57  48  57  19  564*  56 19 


3212 


32*4 


55*.5  54* 


54*7 


33*11' 


3341 


34*13  34*45  35*19  3S*53  36*W  37 


53*S2  52  si  52*i8  5»*4o 


37  4Z  38  » 


oc  5»;<> 

OD  32  o 


57*3! 

32»32~S7 


55  X  55o   5<»  53  54  53  id  52*44 
33*27  33*57'  34*ii  35*0  3S*3i  36*6  36*4i  37*16 


52°8 


Si's 
38*3^39*9 


50V 

39*48 


57  16  5648  56*19  5549  55*  ie  54  47  54*  IB  53  4t  53*6  52  33  5IS7  BIV20  SCT43 


32 44  39  n  3341 


34*,,' 


34V  35*13  35*45j  36*18  36*Sl!  37  27  38*3 


3840 


3917 


504 
39*56 


40*3^ 


37 


SS  35  55  s 

33~t8|33~S6|34%i  34*sb 


6*3^37%' 


50as  4«  S6  49 


373738133846 


39ts40V 


*eV 

40-43  41*23' 


38 


55s. 
3*9 


SSli 


54*si  5423 


35*37  36°  9 


3*42  37 14|  374838s* 


50°27  49*49 
3857  39*3340 


49*> 


39 


3451  3521 


54°ib 
35°5fl 


36°2,' 


S3*V 


52*3 


iTn  50*54  SO*  19  49*41  49°S 


396 


3941 


47*48 
33  42*12 


47*/ 


40 


54*28}S3S8  53*C8  5258  52  to  51° 

36°3237V  37°3438*fe' 


35°32  3fc 


MSI* 


14  3948  40  24  4 


46*24: 

*e°S5  43*36 


11 
41 


5248  52 16 


36  ,i 


36*43  37*2 


51*45  SI    I 

37*441 38*5 


&5039j50s 


49  30  4*  54  48' 


38 48  39  ii  39  55  40  30  41  £ 


17 

41'43 


47*40  47° 


46*12 


42  20  42  (9  43  3» 


51' 
4*19 


50*32  49*! 


484948' 


382*  3866  3928  4O2.|403fc  41  I 


13 

•H*47 


47*36  46°5»  46*20 
42*z4  43*  I 


26 


BROWN    &    SHARPE    MFG.    CO. 


TABLE   3. 

ANGLE  OF  FACE. 


40393837363534333231 


30292827 


12 


7114    IB 

703si70'6' 


IS  t 
€>8« 


16 

66'4« 


17  .a 
fefc*  s 


8  5  I'   1927 

6  3*4  3163' 3 


20  V 

2*9 

27*«4 
0°2« 


5"  17  1 15  3» 

68*47 


6  51 

67*9 


66  33  6556 


8  i* 
65*,6 


18 
64"«4 


I3'«7 

63*5 


20  ail 2 1 

6Z'i4\t>\~i- 


141 


I6°44|6*5S 

60*  oU  7*11 


IB    17 
654, 


1845 

65*a 


9    16 

64-°3i 


2066 
b2°zo 


2/34 
61*42 


60°4i  59*41 


15 


67*. 


19 
64*«6 


20  ii 
63*4* 


20  44 

61°B 


ei  ie 
6  2  »4 


6I°39  60 


23  ID 
60*2 


16 


16*41 
66V 


21's 
63*7 


2I°»4 

6  2°.  e 


6(4.5  6I°0 


53  as  58 


57*o  56^2 


17 


IS  <N 

64 


22«4 

61   50 


22«7 

6I°9 


23»3 

60*t5 


24  i . 


B5> 


59*40  <ra 


£6 14. 
57*.o 


27i927iT 
56*i  sk^"'* 


5-4*,« 

»0*» 

5247 


* 

63V 


2l°a7|22*6 
63*3 


23°4»  24  i. 

60^.i  59°5I 


2614 


26°5T 

56*39 


28\9  29° 

29*S6|304: 


19 


\2Z\o 


2537 


ccw 

S"7°4 


27*3»U8*»« 
55*»t 


tojj 

Fv 

50  *x 


2  fir3  * 
UwV 


23%o  24° i  24°aa  25*6 


28°5 


59*34458*54 


26*5 
54°tf» 


MV 


2IKC 


25°/o|254a 
594659% 


26 
58*11 


574 


31 V 
52*40 


32*4SJ33* 

50*53 


2546  26°9|2653 

|59%« 


27*27| 
57°I9 


28*5 
56*94 


55 


Me 

54*17 


30  4  B 

54* 


33°, 


r»|MVi 

(•asi 


35*54 


r. 

\3bst 


29  14 


56 *»«{5$ao  5*4i  S3<7  53  a 


324* 

i4 


3I°4* 


S55 


33  "14 

Sl'o 


34 
50  °6 


49to 


r. 

156- 


Z9  32* 


3347 
50*5 


r. 

•i  Us 


[30s. 
551, 


5234 


35Y 


504*49 


49  3  48? 


3856 

45°i<J  44~7| 


1 31  3 
54% 


3 

53~37|52«4 


333 


35°3 
494S 


363., 


48*55  48 


38*,6 
46*,o 


40°4|4I°, 
4»*5| 


52*8951*56 


33V7|3439|35» 
50V 


3820 
46% 


59\, 
45°,, 


j32Vi 
L52Y 


39^  402 


51  44]  51 ' 


33«7 
5 1  as 


3436  35  is 
50°so|50°7 


38°7 
«|46*55[46*3 


45% 


40°j 
44*4 


4ltS 


42% 


31 


34-43 

50' 


3652 

48°t( 


37»$ 
4739 


sn 

4652 


395 
4fi*i 


38i/ 
[46" 


45°, 


42  ze 


37°»38°o 


38°4t  39*26  40'0  405 


45 


44*i«  43 


4|'44 

Jfc 


3732  38'..|38*5S 


,40°IB4I04 


45s 


44to 


Sft'tt 
L47°4 


395M40Dt*| 
46°»9|4S*54  45°a| 


4IA5 
43*81 


4241 


139 

46*3Ti 


3962  403441    is 


42. 

14334 


40*0 
\4Sst 


40  *o|  4 1  it  425 
3*»7 


>47 

145*7 


42r 
43V 
42°5al 


4lP 


PROVIDENCE,    R.    I. 


TABLE    ^—{Continued.} 


ANGLE  OF  FACE. 


=  9o°  -  fo  + 
(See  Page  13.) 


28 


BROWN    &    SHARPE   MFG.    CO. 


NATURAL   SINE. 


Deg. 

0' 

10' 

20' 

30' 

40' 

DO'      CO' 

0 

.00000 

.00291 

.00581 

.00872 

.01163 

.01454 

.01745 

89 

1 

.01745 

.02036 

.02326 

.02617 

.02908 

.03199 

.03489 

88 

2 

.03489 

.03780 

.04071 

.04361 

.04652 

.04943 

.05233 

87 

3 

.05233 

.05524 

.05814 

.06104 

.06395 

.06685 

.06975 

1  86 

4 

.06975 

.07265 

.07555 

.07845 

.08135 

.08425 

.08715 

1  85 

5 

.08715 

.09005 

.09295 

.09584 

.09874 

.10163 

.10452 

1  84 

6 

.10452 

.10742 

.11031 

.11320 

.11609 

.11898 

.12186 

83 

7 

.12186 

.12475 

.12764 

.  13052 

.13341 

.  13629 

.13917 

82 

8 

13917 

.14205 

.14493 

.14780 

.  15068 

.15356 

.15643 

81 

9 

15643 

15930 

.16217 

.16504 

.16791 

.17078 

.17364 

80 

10 

.17364 

.17651 

.17937 

.18223 

.  18509 

.18795 

.19080 

79 

11 

.19080 

.19366 

.19651 

.19936 

.20221 

.20506 

.20791 

78 

12 

.20791 

.21075 

.21359 

.21644 

.21927 

.22211 

.22495 

:  77 

13 

.22495 

.22778 

.23061 

.23344 

.23627 

.23909 

.24192 

76 

14 

.24192 

.24474 

.24756 

.25038 

.25319 

.25600 

.25881 

;  75 

15 

.25881 

.26162 

.26443 

.28723 

.27004 

.27284 

.27563 

i  74 

16 

.27563 

.27843 

.28122 

.28401 

.28680 

.28958 

.29237 

73 

17 

.29237 

.29515 

.29793 

.30070 

.30347 

.30624 

.30901 

72 

18 

.30901 

.31178 

.31454 

.31733 

.32006 

.32281 

.32556 

I  71 

19 

.32556 

.32831 

.33106 

.33380 

.33654 

.33928 

.34202 

70 

20 

.34202 

.34475 

.34748 

.35020 

.35293 

.35565 

.35836 

69 

21 

.35836 

.36108 

.36379 

.36650 

.36920 

.37190 

.37460 

68 

22 

.37460 

.37780 

.37909 

.38268 

.38533 

.38805 

.39073 

67 

23 

.39073 

.39340 

.39607 

.39874 

.40141 

.40407 

.40673 

66 

24 

.40673 

.40939 

.41204 

.41469 

.41733 

.41998 

.42261 

65 

25 

.42261 

.42525 

.42788 

.43051 

.43313 

.43575 

.43837 

64 

26 

.43837 

.44098 

.44359 

.44619 

.44879 

.45139 

.45399 

I  63 

27 

.45399 

.45658 

.45916 

.46174 

.46432 

.46690 

.46947 

62 

28 

.46947 

.47203 

.47460 

.47715 

.47971 

.48226 

.48481 

61 

29 

.48481 

.48735 

.48989 

.49242 

.49495 

.49747 

.50000 

I  60 

30 

.50000 

.50251 

.50503 

.50753 

.51004 

.51254 

.51503 

59 

31 

.51503 

.51752 

.52001 

.  52249 

.52497 

.52745 

.52991 

58 

32 

.52991 

.53238 

.53484 

.53730 

.53975 

.54219 

.54463 

57 

33 

.54463 

.54707 

.54950 

.55193 

.55436 

.55677 

.55919 

56 

34 

.55919 

.56160 

.56400 

.56640 

.56880 

.57119 

.57357 

55 

35 

.57357 

.57595 

.57833 

.58070 

.58306 

.58542 

.58778 

54 

36 

.58778 

.59013 

.59248 

.59482 

.59715 

.59948 

.60181 

53 

37 

.60181 

.60413 

.60645 

.60876 

.61106 

.61336 

.61566 

52 

38 

.61566 

.61795 

.62023 

.62251 

.62478 

.62705 

.62932 

51 

39 

.62932 

.63157 

.63383 

.63607 

.63832 

.64055 

.64278 

50 

40 

.64278 

.64501 

.64723 

.64944 

.65165 

.65386 

.65605 

49 

41 

.65605 

.65825 

.66043 

.66262 

.66479 

.66696 

.66913 

48 

42 

.66913 

.67128 

.67344 

.67559 

.67773 

.67986 

.68199 

47 

43 

.68199 

.68412 

.68624 

.68835 

.69046 

.69256 

.69465 

46 

44 

.69465 

.69674 

.69883 

.  70090 

.70298 

.70504 

.70710 

45 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

peg. 

NATURAL   COSINE. 


PROVIDENCE,  R.  I. 


29 


NATURAL   SINE. 


Deg. 

0'       10' 

20' 

30' 

40' 

ft 

GO' 

45 

.70710 

.70916 

.71120 

.71325 

.71528 

.71731 

.71934 

44 

46 

.71934 

.72135 

.72336 

.72537 

.72737 

.  72930 

.73135 

43 

47 

.73135 

.  73333 

.  73530 

.73727 

.73923 

.74119 

.74314 

42 

48 

.74314 

.74508 

.74702 

.74895 

.75088 

.75279 

.75471   41 

49 

.75471 

.  75661 

.75851 

.76040 

.76229 

.76417 

.76604   40 

50 

.76604 

.76791 

.76977 

.77102 

.  77347 

.77531 

.77714 

39 

51 

.77714 

.77897 

.78079 

.78260 

.78441 

.78621 

.78801   38 

52 

.78801 

.78979 

.79157 

.79335 

.79512 

.79688 

.79863   37 

53 

.79863 

.80038 

.80212 

.80385 

.80558 

.80730 

.80901  !  36 

54 

.80901 

.81072 

.81242 

.81411 

.81580 

.81748 

.81915  ;!  35 

55 

.81915 

.82081 

.82247 

.82412 

.82577 

.82740 

.82903   34 

56 

.82903 

.83066 

.83227 

.83383 

.83548 

.83708 

.83867   33 

57 

.83867 

.84025 

.84182 

.84339 

.84495 

.84650 

.84804 

32 

58 

.84804 

.84958 

.85111 

.85264 

.85415 

.85566 

.85716 

31 

59 

.85716 

.85866 

.86014 

.86162 

.88310 

.86456 

.86602 

30 

60 

.86602 

.86747 

.86892 

.87035 

.87178 

.87320 

.87462 

29 

61 

.87462 

.87602 

.87742 

.87881 

.88020 

.88157 

.88294 

28 

62 

.88294 

.88430 

.88566 

.88701 

.88835 

.88968 

.89100 

27 

C3 

.89100 

.89232 

.89363 

.89493 

.89622 

.89751 

.89879   26 

64 

.89879 

.90006 

.90132 

.90258 

.90383 

.90507 

.90630   25 

65 

.90630 

.90753 

.90875 

.90996 

.91116 

.91235 

.91354   24 

G6 

.91354 

.91472 

.91580 

.91706 

.91821 

.91936 

.92050   23 

67 

.92050 

.92163 

.92270 

.92388 

.92498 

.92609 

.92718   22 

68 

.92718 

.92827 

.92934 

.93041 

.93148 

.93253 

.93358   21 

69 

.93358 

.93461 

.93565 

.93667 

.93768 

.93869 

.93969   20 

70 

.93969 

.94068 

.94166 

.94264 

.94360 

.94456 

.94551  '  19 

71 

.94551 

.94646 

.94739 

.94832 

.94924 

.95015 

.95105  !  18 

72 

.95105 

.95195 

.95283 

.95371 

.95458 

.95545 

.95630   17 

73 

.95630 

.95715 

.95799 

.95882 

.95964 

.96045 

.96126   16 

74 

.96126 

.96205 

.96284 

.96363 

.96440 

.96516 

.96592   15 

75 

.96592 

.96667 

.96741 

.96814 

.96887 

.96958 

.97029   14 

76 

.97029 

.97099 

.97168 

.97237 

.97304 

.97371 

.97437  ;  13 

77 

.97437 

.97502 

.97566 

.9762!) 

.97692 

.97753 

.97814   12 

78 

.97814 

.97874 

.97934 

.97992 

.98050 

.98106 

.98162  1  11 

79 

.98162 

.98217 

.98272 

.98325 

.98378 

.98429 

.98480  !  10 

80 

.98480 

.98530 

.98580 

.98628 

.98670 

.  98722 

.98768 

9 

81 

.98768 

.98813 

.98858 

.98901 

.98944 

.98985 

.99026 

8 

82 

.99026 

.99066 

.09106 

.99144 

.99182 

.99218 

.99254  ! 

7 

83 

.99254 

.99289 

.99323 

.99357 

.99389 

.99421 

.99452 

6 

84 

.99452 

.99482 

.99511 

.99539 

.99567 

.99593 

.99619 

5 

85 

.99619 

.99644 

.99668 

.99691 

.99714 

.99735 

.99756 

4 

86 

.99756 

.99776 

.99795 

99813 

.99830 

.99847 

.99863 

3 

87 

.99863 

.99877 

.99891 

.99904 

.99917 

.99928 

.99939 

2 

88 

.99939 

.99948 

.99957 

.99965 

.99972 

.99979 

.99984 

1 

89 

.99984 

.99989 

.99993 

.99996 

.99998 

.99999 

1.0000 

0 

- 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL   COSINE. 


3° 


BROWN    &    SHARPE    MFG.    CO. 


NATURAL  TANGENT. 


J*«. 

0' 

10' 

£0' 

30' 

40' 

50' 

60' 

0 

.00000 

.00290 

.00581 

.00872 

.01163 

.01454 

.01745 

89 

1 

.01745 

.02036 

.02327 

.02618 

.02909 

.03200 

.03492 

88 

2 

.03492 

.03783 

.04074 

.04366 

.04657 

.04949 

.05240 

87 

3 

.05240 

.05532 

.05824 

.06116 

.06408 

.06700 

.06992 

86 

4 

.06992 

.07285 

.07577 

.07870 

.08162 

.08455 

.08748 

85 

5 

.08748 

.09042 

.09335 

.09628 

.09922 

.10216 

.10510 

84 

6 

.10510 

.10804 

.11099 

.11393 

.11688 

.11983 

.12278 

83 

7 

.12278 

.12573 

.  12869 

.13165 

.13461 

.  13757 

.14054 

82 

8 

.14054 

.14350 

.14647 

.14945 

.15242 

.15540 

.15838 

81 

9 

.15838 

.16136 

.16435 

.16734 

.17033 

.17332 

.17632 

80 

10 

.17632 

.17932 

.18233 

.18533 

.18834 

.19136 

.19438 

79 

11 

.19438 

.  19740 

.20042 

.20345 

.20648 

.20951 

.21255 

78 

12 

.21255 

.21559 

.21864 

.22169 

.22474 

.22780 

.23086 

77 

13 

.23086 

.23393 

.23700 

.24007 

.24315 

.24624 

.24932 

76 

14 

.24932 

.25242 

.25551 

.25861 

.26172 

.26483 

.26794 

75  < 

15 

.26794 

.27106 

.27419 

.27732 

.28046 

.28360 

.28674 

74 

10 

.28674 

.28989 

.29305 

.29621 

.29938 

.30255 

.30573 

73 

17 

.30573 

.30891 

.31210 

.31529 

.31850 

.32170 

.32492 

72 

18 

.32492 

.32813 

.33136 

.33459 

.33783 

.34107 

.34432 

71 

19  ; 

.34432 

.34753 

.35084 

.35411 

.35739 

.36067 

.36397 

70 

20  i 

.36397 

.36726 

.37057 

.37388 

.37720 

.38053 

.38386 

69 

21 

.38386 

.38720 

.39055 

.39391 

.39727 

.40064 

.40402 

:  68 

22 

.40402 

.40741 

.41080 

.41421 

.41762 

.42104 

.42447 

|  67 

23 

.42447 

.42791 

.43135 

.434S1 

.43827 

.44174 

.44522 

i  66 

24 

.44522 

.44871 

.45221 

.45572 

.45924 

.46277 

.46630 

65 

25  | 

.46630 

.46985 

.47341 

.47697 

.48055 

.48413 

.48773 

64 

26 

.48773 

.49133 

.49495 

.49858 

.50221 

.50586 

.50952 

63 

27 

.50952 

.51319 

.51687 

.52056 

.52427 

.52798 

.53170 

62 

28 

.53170 

.53544 

.53919 

.54295 

.54672 

.55051 

.55430 

61 

29  ' 

.55430 

.55811 

.56193 

.56577 

.56961 

.57847 

.57735 

60 

30 

.57735 

.58123 

.58513 

.58904 

.59297 

.59690 

.60086 

59 

31  ! 

.60086 

.60482 

.60880 

.61280 

.61680 

.62083 

.62486 

58 

32 

.62486 

.62892 

.63298 

.63707 

.64116 

.64528 

.64940 

57 

33 

.64940 

.65355 

.65771 

.66188 

.66607 

.67028 

.67450 

56 

34 

.67450 

.67874 

.68300 

.68728 

.69157 

.69588 

.70020 

55 

35 

.70020 

.70455 

.70891 

.71329 

.71769 

.72210 

.72654 

54 

30 

.72654 

.73099 

.73546 

.73996 

.74447 

.74900 

.75355 

53 

37 

.  75355 

.  75812 

.76271 

.76732 

.77195 

.77661 

.78128 

52 

38 

.78128 

.78598 

.79069 

.79543 

.80019 

.80497 

80978 

51 

39 

.80978 

.81461 

.81946 

.82433 

.82923 

.83415 

.83910 

50 

40 

i  .83910 

.84406 

.84906 

.85408 

.85912 

.86419 

.86928 

49 

41 

.86928 

.87440 

.87955 

.88472 

.88992 

.89515 

.90040 

48 

42 

.90040 

.90568 

.91099 

.91633 

.92169 

.92709 

.93251 

47 

43 

.93251 

.93796 

.94345 

.94896 

.95450 

.96008 

.96568 

46 

44 

.96568 

.97132 

.97699 

.98269 

.98843 

.99419 

1.0000 

45 

60' 

50' 

40' 

30 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENT. 


PROVIDENCE,  R.  I. 


NATURAL  TANGENT. 


Deg. 

0' 

10' 

20' 

SO' 

40' 

50' 

60 

45 

1.0000 

1.0058 

1.0117 

1.0176 

1.0235 

.0295 

1.0355 

44 

46 

1.0355 

1.0415 

1.0476 

1.0537 

1.0599 

.0661 

1.0723 

43 

47 

1.0723 

1.0786 

1.0849 

1.0913 

1.0977 

.1041 

1.1106 

42 

48 

1.1106 

1.1171 

1.1236 

1.1302 

1  .  1369 

.1436 

1.1503 

41 

49 

1.1503 

1.1571 

1.1639 

1.1708 

1.1777 

.1847 

1.1917 

40 

50 

1.1917 

1.1988 

1.2059 

1.2131 

1.2203 

.2275 

1.2349 

39 

51 

1.2349 

1.2422 

1.2496 

1.2571 

1.2647 

.2723 

1.2799 

38 

52 

1.2799 

1.2876 

1.2954 

1.3032 

1.8111 

.3190 

1.3270 

37 

53 

1.3270 

1.3351 

1.3432 

1.3514 

.3596 

.3680 

1.3763 

36 

54 

1.3763 

1.3848 

1.3933 

1.4019 

.4106 

.4193 

1.4281 

35 

55 

1.4281 

1.4370 

1.4459 

1.4550 

.4641 

.4733 

1.4825 

34 

56 

1.4825 

1.4919 

1.5013 

1.5108 

.5204 

.5301 

1.5398 

33 

57 

1.5398 

1.5497 

1.5596 

1.5696 

.5798 

.5900 

1.6003 

32 

58 

1.6003 

1.6107 

1.6212 

1.6318 

.6425 

.6533 

1.6642 

31 

59 

1.6642 

1.6753 

1.6864 

1.6976 

.7090 

.7204 

1.7320 

30 

60 

1.7320 

1.7437 

1.7555 

1  .  7674 

.7795 

1.7917 

1.8040 

29 

61 

1.8040 

1.8164 

1.8290 

1.8417 

1.8546 

1.8676 

1.8807 

28 

62 

1.8807 

1.8940 

1.9074 

1.9209 

1.9347 

1.9485 

1.9626 

27 

63 

1.9626 

1.9768 

1.9911 

2.0056 

2.0203 

2.0352 

2.0503 

26 

64 

2.0503 

2.0655 

2.0809 

2.0965 

2.1123 

2.1283 

2.1445 

25 

65 

2.1445 

2.1609 

2.1774 

2.1943 

2.2113 

2.2285 

2.2460 

24 

66 

2.2460 

2.2637 

2.2816 

2.2998 

2.3182 

2.3369 

2.3558 

23 

67 

2.3558 

2.3750 

2.3944 

2.4142 

2.4342 

2.4545 

2.4750 

22 

68 

2.4750 

2.4959 

2.5171 

2.5386 

2.5604 

2.5826 

2.6050 

21 

69 

2.6050 

2.6279 

2.6510 

2.6746 

2.6985 

2.7228 

2.7474 

20 

70 

2.7474 

2.  '7725 

2.7980 

2.8239 

2.8502 

2.8770 

2.9042 

19 

71 

2.9042 

2.9318 

2.9600 

2.9886 

3.0178 

3.0474 

3.0776 

18 

72 

3.0776 

3.1084 

3.1397 

3.1715 

3.2040 

3.2371 

3.2708 

17 

73 

3.2708 

3.3052 

3.3402 

3.3759 

3.4123 

3.4495 

3.4874 

16 

74 

3.4874 

3.5260 

3.5655 

3.6058 

3.6470 

3.6890 

3.7320 

15 

75 

3.7320 

3.7759 

3.8208 

3.8667 

3.9136 

3.9616 

4.0107 

14 

76 

!  4.0107 

4.0610 

4.1125 

4.1653 

4.2193 

4.2747 

4.3314 

13 

77 

!  4.3314 

4.3896 

4.4494 

4.5107 

4.5736 

4.6382 

4.7046 

12 

78 

!  4.7046 

4.7728 

4.8430 

4.9151 

4.9894 

5.  0658 

5.1445 

11 

79 

|  5.1445 

5.2256 

5.3092 

5.3955 

5.4845 

5.5763 

5.6712 

10 

80 

5.6712 

5.7693 

5.8708 

5.9757 

6.0844 

6.1970 

6.3137 

9 

81 

|  6.3137 

6.4348 

6.5605 

6.6911 

6.8269 

6.9682 

7.1153 

8 

82 

7.1153 

7.2687 

7.4287 

7.5957 

7.7703 

7.9530 

8.1443 

7 

83 

1  S.1443 

8.3449 

8.5555 

8.7768 

9.0098 

9.2553 

9.5143 

6 

84 

9.5143 

9.7881 

10.078 

10.385 

10.711 

11.059 

11.430 

5 

85 

11.430 

11.826 

12.250 

12.706 

13.196 

13.726 

14.300 

4 

86 

14.300 

14.924 

15.604 

16.349 

17.169 

18.075 

19.081 

3 

87 

19.081 

20.205 

21.470 

22.904 

24.541 

26.431 

28.636 

2 

88 

28.636 

31.241 

34.367 

38.188 

42.964 

49.103 

57.290 

1 

89 

57.290 

68.750 

85.939 

114.58 

171.88 

343.77 

00 

0 

60 

50 

40* 

30' 

20'  . 

10' 

0' 

Deg. 

NATURAL  COTANGENT. 


BROWN    &    SHARPE    MFG.    CO. 


IV. 


WORM   AND  WORM  WHEEL 

(Fig.  8.) 


PROVIDENCE,    R.    I.  33 


FORMULAS. 

L  =  lead  of  worm. 

N  =  number  of  teeth  in  gear. 

m  —  threads  per  inch  in  worm. 

d=  diameter  of  worm. 
d'  —  diameter  of  hob. 
T  =  throat  diameter. 
B  =  blank  diameter  (to  sharp  corners). 
C  =  distance  between  centers. 

o  =  thickness  of  hob-slotting  cutter. 

/=  width  of  bands  at  bottom. 

b  =  pitch  circumference  of  worm. 

v  =  width  of  worm  thread  tool  at  end. 
w  =  width  of  worm  thread  at  tap. 
P  =  diametral  pitch. 
P1  =  circular  pitch. 

s  =  addendum. 

t  —  thickness  of  tooth  at  pitch  line. 
tn  =  normal  thickness  of  tooth. 
/=  clearance  at  bottom  of  tooth. 
D"  =  working  depth  of  tooth. 
D"  +  /  =  whole  depth  of  tooth. 

6  =  angle  of  thread  with  axis. 

If  the  lead  is  for  single,  double,  triple,  etc.,  thread,  then 
L=P',  2P',3P',etc. 


U&I7EESIT7 


34  BROWN    &    SHARPE    MFG.    CO. 


a  =  60°  to  90° 
m 

p,=        7TT 

N    +    2 

D  =  NP;  =  N 

n          P 


I  =  n  (d  -  2  S) 

,  »  _  L       j  Practical  only  when  width  of  wheel  on  wheel  pitch  circle 

~  £         (          is  not  more  than  %  pitch  diameter  of  worm. 


i      d 
r  —  -  —  2  s 

2 

r*  =  r'  +  D"  +  / 
D  +  </ 


B  -  T  +  2  (r1  -  r1  cos  ?)       A  mea^rement  of 

\  2  /  sufficient. 


sketch  is  generally 


d'**d+  *f 

™=  -335  p' 

NOTE. — The  notations  and  formulas  referring  to  tooth  parts,  given  on  page  5  for 
spur  gears,  apply  to  worm  wheels,  and  are  here  used. 

NOTE. — Hob  and  worm  should  be  marked,  as  per  example  : 
4  threads  per  i"  single  .25  P';  .25  L. 
2  threads  per  i"  double  .25  P';  .50  L. 


PROVIDENCE,    R.    I. 


UNDERCUT  IN  WORM  WHEELS. 


35 


In  worm  wheels  of  less  than  30  teeth  the  thread  of  the  worm 
(being  29°)  interferes  with  the  flank  of  the  gear  tooth.  Such 
a  wheel  finished  with  a  hob  will  have  its  teeth  undercut.  To 
avoid  this  interference  two  methods  may  be  employed. 

First  Method.  —  Make  throat  diameter  of  wheel 


=  cos 


or 


This  formula  increases  the  throat  diameter,  and  conse- 
quently the  center  distance.  The  amount  of  the  increase  can 
be  found  by  comparing  this  value  of  T  with  the  one  as  obtained 
by  formula  on  page  34.  To  keep  the  original  center  distance, 
the  outside  diameter  of  the  worm  must  be  reduced  by  the 
same  amount  the  throat  diameter  is  increased. 

Second  Method. — Without  changing  any  of  the  dimensions 
we  found  by  the  formulas  given  on  page  34,  we  can  avoid  the 
interference  to  be  found  in  worm  wheels  of  less  than  30  teeth 
by  simply  increasing  the  angle  of  worm  thread.  We  find  the 
value  of  this  angle  by  the  following  formula  : 
Let  there  be 

2  Y  ~  angle  of  worm  thread. 

N  =  number  of  teeth  in  worm  wheel. 

cos  Y 
From  this  formula  we  obtain  the  following  values  : 


N 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

2  y 

3oK 

3i 

31/2 

32^ 

32^ 

33^ 

34K 

35 

36 

37 

N 

19 

18 

17 

16 

15 

14 

i3 

12 

2  y 

1  38 

39 

40 

4i  j* 

42^ 

44^ 

46^ 

48 

As  this  latter  formula  involves  the  making  of  new  hobs  in 
many  cases,  on  account  of  change  of  angle,  we  prefer  to  reduce 
the  diameter  of  worm  as  indicated  by  first  method,  if  the  dis- 
tance of  centers  must  be  absolute. 


BROWN    &   SHARPE    MFG.    CO. 


CHARTER  v. 


SPIRAL  OR  SCREW  GEARING. 

(Figs.  9,  10,  ii.) 


Fig.  9. 

In  spiral  gearing  the  wheels  have  cylindrical  pitch  surfaces, 
but  the  teeth  are  not  parallel  to  the  axis.  The  line  in  which 
the  pitch  surface  intersects  the  face  of  a  tooth  is  part  of  a 
screw  line,  or  helix,  drawn  at  the  pitch  surface.  A  screw 
wheel  may  have  one  or  any  number  of  teeth.  A  one-toothed 
wheel  corresponds  to  a  one-threaded  screw,  a  many-toothed 
wheel  to  a  many- threaded  screw.  The  axes  may  be  placed  at 
any  angle. 

Consider  spiral  gears  with  : 

I.  Axes  parallel. 
II.  Axes  at  right  angles. 
III.  Axes  any  angle. 


PROVIDENCE,    R.    I. 


37 


Fig.  10. 

Let  there  be : 

XT"  —    r  number  of  teeth  in  gears  •!    ? 

Nb  -  )  (   b  v „ 

C  =  center  distance. 
P  =  diametral  pitch 
P'  =  circular  pitch. 
Pn  =  normal  diametral  pitch. 
P'n  =  normal  circular  pitch. 
y  =  angle  of  axes. 

Lj  =  exact  lead  of  spiral  on  pitch  surface. 
L2  =  approximate  lead  of  spiral  on  pitch  surface. 
T  —  number  of  teeth  marked  on  cutter  to  be  used  when 

teeth  are  to  be  cut  on  milling  machine. 
D  =  pitch  diameter. 
B  =  blank  diameter. 

a  Z    [  angle  of  teeth  with  axis 

6  J 

/  =  thickness  of  tooth. 
s  =  addendum. 
D"  +  /  =  whole  depth  of  tooth. 

NOTE. — Letters  a  and  b  occurring  at  bottom  of  notations  refer  to  gears  a  and  b, 

I. — AXES  PARALLEL. 

Gears  of  this  class  are  called  twisted  gears.  The  angle  of 
teeth  with  axes  in  both  gears  must  be  equal  and  the  spirals 
run  in  opposite  directions.  The  angles  are  generally  chosen 
small  (seldom  over  20°)  to  avoid  excessive  end  thrust.  End 
thrust  may,  however,  be  entirely  avoided  by  combining  two 
pairs  of  wheels  with  right  and  left-hand  obliquity.  Gears  of 
this  class  are  known  as  Herringbone  gears.  They  are  com- 
paratively noiseless  running  at  high  speed. 


38  BROWN    &    SHARPE    MFG.    CO. 

II.  —  AXES  AT  RIGHT  ANGLES. 
Here  we  must  always  have  : 

1.  The  teeth  of  same  hand  spiral  ; 

2.  The  normal  pitches  equal  in  both  gears  ;  and 

3.  The  sum  of  the  angles  of  teeth  with  axes  =  90°. 

CHOOSING  ANGLE  OF  TEETH  WITH  AXES. 

1.  If  in  a  pair  of  gears  the  ratio  of  the  number  of  teeth  is 
equal  to  the  direct  ratio  of  the  diameters,  /.  <?.,  if  the  number  of 
teeth  in  the  two  gears  are  to  each  other  as-  their  pitch  diame- 
ters, then  the  angles  of  the  spirals  will  be  45°  and  45°  ;  for,  this 
condition  being  fulfilled,  the  circular  pitches  of  the  two  gears 
must  be  alike,  which  is  only  possible  with  angles  of  45°.     In 
such  a  combination  either  gear  may  be  the  driver. 

2.  If  the  ratio  of  the  diameters  determined  upon  is  larger 
or  smaller  than  the  ratio  of  the  number  of  teeth,   then  the 
angles  are  : 


In  such  gears  the  velocity  ratio  is  measured  by  the  number 
of  teeth,  and  not  by  the  diameters. 
3.  Given  Na,  Nb  and  C  : 
If  P'  is  made  =  P'  then  we  have  case  "  i  "  and 


p,  _ 

~ 


But  if  Pa'  is  assumed,  then  : 


_ 

/sN, 
and 

tan  aa  =  ^L         tan  ab  =  ?± 

P&  Pa 

The  gear  whose  P'  or  a  is  larger  will  be  the  driver,  on 
account  of  the  greater  obliquity  of  the  teeth. 
4.  Given  Na,  N&  and  C  or  D. 
See  case  "  7  "  under  III.,  considering  ;/  =  90°. 

III.  —  Axis  AT  ANY  ANGLE  (y). 

5.  Given  case  "  i,"  under  II.,  then  angles  of  spirals  =  %  y, 
for  the  same  reason. 

6.  Analogous  cases  to  "2"  and  "3,"  under  II.,   may   be 
worked  out,  when  angles  of  axes  =  y,  but   they   have   been 


PROVIDENCE,    R.    I. 


39 


omitted,  partly  because  the  formulas  are  too  cumbersome,  and 
partly  because  they  are  to  some  extent  covered  by  cases  "  5  " 
and  "7." 

7.  Given  N0,  N6  and  C,  or  one  of  the  pitch  diameters.  We 
find  the  angles  by  a  graphic  method,  which  for  all  practical 
purposes  is  accurate  enough  ;  ro  and  v  o  are  the  axes  of  gears 
forming  angle  y  (see  diagram,  Fig.  n.)  On  these  axes  we 
lay  off  lines  o  r  and  o  v  representing  the  ratio  of  the  number 
of  teeth  (velocity  ratio),  so  that  Na  :  N6  :  :  r  s  :  s  v,  and 


construct  parallelogram  o  r  s  v.  Then,  according  to  Mc- 
Cord,*  the  angles  formed  by  the  tangent  s  o  in  the  pitch  con- 
tafet  o  with  the  axes  of  the  gears  insures  the  least  amount  oj 
sliding.  In  bisecting  angle  y  by  tangent  u  o  and  using  angles 
produced  in  this  manner  we  equally  distribute  the  end  thrust  on 
both  shafts.  Both  methods  have  their  advantages  ;  to  profit 
by  both  we  select  angles  aa  and  ab,  produced  by  tangent  o  x9 
bisecting  angle  u  o  s. 

Thus  we  have  when  angles  are  found  and  C  given, 


Ptn  *  ^ 

z^    -  •  — ' — 


2  C  7t  cos  an  cos 


Nn  cos  ofb  -f 
ind  when  Da  given 
p,B      Da  n  cos  aa 

N6  COS  £Ta 

and 

Na 
D    -    P'nN<> 

7t  COS  ^6 

*  McCord,  Kinematics,  page  378. 


40  BROWN  &  SHARPE  MFG.  CO. 

GENERAL   FORMULAS. 

p   fn  p  in 

*•  a       —  -1  6 

P'nN 


7t  7t  COS  a 

=  D  +  2J      or  =  D 


P"' 


P'=^     or  = 

N  cos  a 

P'n  =  P'  cos  a 

Pn=  JL     (Pitch  of  cutter.) 

j-?!!     or~  — 

7t  ~   Pn 


=2S   +   - 
10 


(See  Note  I.} 


or  = or  = 


tan  #  P  tan  <*  tan  a  cos  # 


/cos    45°  =  .707ii\ 
Icos'  45°  =  -So       / 


NOTE  i. — Cutters  of  regular  involute  system. 


Use  No.  i  cutter  for  T  from      135  up. 


"    "     " 


55  to  134 
35  to  54 
26  to  34 


No.  5  cutter  for  T  from  21  to  25 

«     5      tt        ci     «c      i«  I7  to  20 

ti     ?      it        ti     it      «  I4  to  l6 

u       g        u          (i      <i        ««  I2  to  x, 


Note  2. — Gears  used  on  spiral  head  and  bed   for   Brown   &   Sharpe  milling 
machine  : 

W  =  number  of  teeth  in      gear  on  worm. 
Gi  =  "  "      ist   "          stud. 

Gz  =  "  "      2d    "          stud. 

S  =  "  "  "          screw. 

Should  a  spiral  head  of  different  construction  be  used,  the  formula  would  not 
apply. 


PROVIDENCE,    R.    I.  41 


VI. 


INTERNAL  GEARING. 

PART  A.—  INTERNAL  SPUR  GEARING. 
(Figs.  12,  13,  14,  15,  16.) 

A  little  consideration  will  show  that  a  tooth  of  an  internal 
or  annular  gear  is  the  same  as  the  space  of  a  spur  —  external 
gear. 

We  prefer  the  epicycloidal  form  of  tooth  in  this  class  of 
gearing  to  the  involute  form,  for  the  reason  that  the  difficulties 
in  overcoming  the  interference  of  gear  teeth  in  the  involute 
system  are  considerable.  Special  constructions  are  required 
when  the  difference  between  the  number  of  teeth  in  gear  and 
pinion  is  small. 

In  using  the  system  of  epicycloidal  form  of  tooth  in  which 
the  gear  of  15  teeth  has  radial  flanks,  this  difference  must  be 
at  least  15  teeth,  if  the  teeth  have  both  faces  and  flanks.  Gears 
fulfilling  this  condition  present  no  difficulties.  Their  pitch 
diameters  are  found  as  in  regular  spur  gears,  and  the  inside 
diameter  is  equal  to  the  pitch  diameter,  less  twice  the  adden- 
dum. 

If,  however,  this  difference  is  less  than  15,  say  6,  or  2,  or  i, 
then  we  may  construct  the  tooth  outline  (based  on  the  epicy- 
cloidal system)  in  two  different  ways. 

First  Method.  —  To  explain  this  method  better,  let  us  sup- 
pose the  case  as  in  Fig.  12,  in  which  the  difference  between 
gear  and  pinion  is  more  than  15  teeth.  Here  the  point  o  of 
the  describing  circle  B  (the  diameter  of  which  in  the  best 
practice  of  the  present  day  is  equal  to  the  pitch  radius  of  a  15 
tooth  gear,  of  the  same  pitch  as  the  gears  in  question)  gene- 
rates the  cycloid  o,  o1,  o2,  o3,  etc.,  when  rolling  on  pitch  circle 
L  L  of  gear,  forming  the  face  of  tooth  ;  and  when  rolling  on 
the  outside  of  L  L  the  flank  of  the  tooth.  In  like  manner  is  the 
face  and  flank  of  the  pinion  tooth  produced  by  B  rolling  out- 
side and  inside  of  E  E  (pitch  circle  of  pinion).  A  little  study 


BROWN    &    SHARPE    MFG.    CO. 


of  Fig.   12    (in   which  the  face  and   flank  of  a  gear  tooth  are 
produced)  will   show  the  describing  circle  B  divided  into  12 


equal  parts  and  circles  laid  through  these  points  (i,  2,  3,  etc.), 
concentric  with  L  L.  We  now  lay  off  on  L  L  the  distances 
o-i,  1-2,  2-3,  etc.,  of  the  circumference  of  B,  and  obtain  points 


PROVIDENCK,    R.    I. 


43 


i1,  21,  31,  etc.  [Ordinarily  it  is  sufficient  to  use  the  chord.]  It 
will  now  readily  be  seen  that  B  in  rolling  on  L  L  will  success- 
ively come  in  contact  with  i1,  21,  3',  etc.,  c  meanwhile  moving 
to  c\  f,  c3,  etc.  (points  on  radii  through  i1,  2l,  3*,  etc.),  and  the 
generating  point  o  advancing  to  o1,  o2,  o3,  etc.,  being  the  inter- 
sections of  B  with  c\  c\  c\  etc.,  as  centers  and  the  circles  laid 
through  i,  2,  3,  etc.  Points  o,  o1,  o\  o3,  etc.,  connected  with  a 
curve  give  the  face  of  the  tooth  ;  in  like  manner  the  flank  is 
obtained. 

In  this  manner  the  form  of  tooth  is  obtained,  when  the 
difference  of  teeth  in  gear  and  pinion  is  less  than  15,  with  the 
exception  that  the  diameter  of  describing  circle  B 


where  P  =  diametral  pitch,  N  and  n  number  of  teeth  in  gears. 
The  distances  of  the  tooth  above  and  below  the  pitch  line 
as  well  as  the  thickness  /  are  determined  as  in  regular  spur 
gears  by  the  pitch,  except  when  the  difference  in  gear  and 
pinion  is  very  small,  where  we  obtain  a  short  tooth,  as  in  Figs. 
13  and  14.  In  such  a  case  the  height  of  tooth  is  arbitrary  and 
only  conditioned  by  the  curve.  In  internal  gears  it  is  best  to 
allow  more  clearance  at  bottom  of  tooth  than  in  ordinary  spur 
gears. 


42  T. 


3O  Teeth 


Fig.  13. 


In  a  construction  of  this  kind  it  is  suggested  to  draw  the 
tooth  outline  many  times  full  size  and  reduce  by  photography. 
An  equally  multiplied  line  A  B  will  help  in  reducing. 


44 


BROWN    &    SHARPE    MFG.    CO. 


PROVIDENCE,    R.    I.  45 

Second  Method, — The  difference  between  gear  and  pinion 
being  very  small,  it  is  sometimes  desirable  to  obtain  a  smooth 
action  by  avoiding  what  is  termed  the  "  frict'on  of  approach- 
ing action."*  This  is  done,  the  pinion  driving,  by  giving  gear 
only  flanks,  Fig.  15,  and  the  gear  driving,  by  giving  gear  only 
faces,  Fig.  16.  In  both  these  cases  we  have  but  one  describ- 
ing circle,  whose  diameter  is  equal  to  the  difference  of  the  two 
pitch  diameters.  The  construction  of  the  curve  is  precisely 
the  same  as  described  under  A.  The  describing  circle  has 
been  divided  into  24  parts  simply  for  the  sake  of  greater 
accuracy. 


PART  B.—  INTERNAL  BEVEL  GEARS. 


The  pitch  surfaces  of  bevel  gears  are  cones  whose  apexes 
are  at  a  common  point,  rolling  upon  each  other.  The  tooth 
forms  for  any  given  pair  of  bevel  gears  are  the  same  as  for  a 
pair  of  spur  gears  (of  same  pitch)  whose  pitch  radii  are  equal  to 
the  respective  apex  distances  of  the  normal  cones  (/'.  e.,  cones 
whose  elements  are  perpendicular  upon  the  elements  of  the 
bevel  gear  pitch  cones).  (Compare  Fig  19,  page  50.) 

The  same  is  true  of  internal  bevel  gears,  with  the  modifica- 
tion that  here  one  of  the  pitch  cones  rolls  inside  of  the  other. 
The  spur  gears  to  whose  tooth  forms  the  forms  of  the  bevel 
gear  teeth  correspond,  resolve  themselves  into  internal  spur 
gears  (Fig.  17).  The  problem  is  now  to  be  solved  as  indicated 
in  the  first  part  of  this  chapter. 


*  McCord,  Kinematics,  pages  107,  108. 


46 


BROWN    &    SHARPE    MFG.    CO. 


8  f. 

<;<-<t  i-    40  Tt-cth 
fluion    30 


Fig.  17. 


PROVIDENCE,    R.    I.  47 


CHAPTER  vn. 
GEAR    PATTERNS. 

(Fig.  18.) 

To  place  in  bevel  gears  the  best  iron  where  it  belongs,  the 
tooth  side  of  the  pattern  should  always  be  in  the  nowel,  no 
matter  of  what  shape  the  hubs  are. 

Hubs,  if  short,  may  be  left  solid  on  web  ;  if  long  they  should 
be  made  loose.  A  long  hub  should  go  on  a  tapering  arbor,  to 
prevent  tipping  in  the  sand.  i°  taper  for  draft  on  hubs  when 
loose,  and  3°  when  solid  is  considered  sufficient. 

Coreprints  as  a  rule  are  made  separate,  partly  to  allow  the 
pattern  to  be  turned  on  an  arbor,  partly  for  convenience, 
should  it  be  desirable  to  use  different  sizes. 

Put  rap-  and  draw-holes  as  near  to  center  as  possible. 
Referring  to  Fig.  18,  make  L  =  D  for  D  from  ^"  to  i/^",  or 
even  more,  should  hubs  be  very  long.  Otherwise  if  D  is  more 
than  \y>z"  leave  L  =  i^£". 

Iron  pattern  before  using  should  be  marked,  rusted  and 
waxed. 

Shrinkage  —  For  cast-iron,  "ft"  per  foot. 
For  brass,        T3^"        " 

Cast-iron  gears,  especially  arm  gears,  do  not  shrink  T/8"  per 
foot.  In  making  iron  patterns  the  following  suggestions  have 
been  found  useful  : 

Up  to  12"  diameter  allow  no  shrink. 
From  12"  to  18"  "      y?>  regular  shrink. 

"    i  s"  to  24"      w         "    y2     " 

"      24"  to  48"        "  "     YZ      "  " 

Above  48"         "  "     .10" 

for  cast-iron. 


CJHI7EKSITY 


48 


BROWN    &    SHARPE    MFG.    CO. 


PROVIDENCE,    R.    I. 


49 


If  in  gears  the  teeth  are  to  be  cast,  the  tooth  thickness  /  in 
the  pattern  is  made  smaller  than  called  for  by  the  pitch,  to  avoid 
binding  of  the  teeth  when  cast.  No  definite  rule  can  be  given, 
as  the  practice  varies  on  this  point.  For  the  different  diam- 
etral pitches  we  would  advise  making  /  smaller  by  an  amount 
expressed  in  inches,  as  given  in  the  following  table  : 


DIAM.  PITCH. 

AMOUNT  t 
is  SMALLER. 

DIAM.  PITCH. 

AMOUNT  / 
is  SMALLER. 

16 

.010" 

5 

.020" 

12 

012" 

4 

.022" 

10 

.014" 

3 

.026" 

8 

.016" 

2 

.030" 

6 

.018" 

I 

.040" 

50  BROWN    &    SHARPE    MFG.    CO. 


CHAPTER  viu. 

DIMENSIONS   AND    FORM    FOR    BEVEL    GEAR 

CUTTERS. 

(Fig.   19.) 

The  data  needed  to  determine  the  form  and  thickness  of  a 
bevel  gear  cutter  are  the  following  : 
P  =  pitch. 

N  =  number  of  teeth  in  large  gear. 
n=  number  of  teeth  in  small  gear. 
F  =  length  of  face  of  tooth,  measured  on  pitch  line. 
After  having  laid  out  a  diagram  of  the  pitch  cones  a  b  c  and 
a  b  f,  and  laid  off  the  width  of  face,  the  problem  resolves  itself 
into  two  parts  : 

PART  I.  —  DETERMINE  PROPER  CURVE  FOR  CUTTER. 
It  will  be  remembered  that  in  the  involute  system  of  cutters 
(the  only  one  used  for  bevel  gears  that  are  cut  with  rotary 
cutter),  a  set  of  eight  different  cutters  is  made  for  each 
pitch,  numbering  from  No.  i  to  No.  8,  and  cutting  from 
a  rack  to  12  teeth.  Each  number  represents  the  form  of 
a  cutter  suitable  to  cut  the.  indicated  number  of  teeth.  For 
instance,  No.  4  cutter  (No.  4  curve)  will  cut  26  to  34  teeth. 
In  order  to  find  the  curve  to  be  used  for  gear  and  pinion 
we  simply  construct  the  normal  pitch  cones  by  erecting 
the  perpendicular  p  q  through  $,  Fig.  19.  We  now  measure  the 
lines  b  q  and  b  p,  and  taking  them  as  radii,  multiplying  each  by 
2  and  P  we  obtain  a  number  of  teeth  for  which  cutters  of 
proper  curves  may  be  selected.  From  example  we  have  : 

Gear  :     b  q  —  9^"  ;   2  X  P  X  9.75  =  97  T       No.  2  curve. 

Pinion:  b  p  =  3/2"  ;  2  X  P  X  3.5    =  35  T      No-  3  curve. 
The  eight  cutters  which  are  made  in  the  involute  system 
for  each  pitch  are  as  follows  : 

No.  i  will  cut  wheels  from  135  teeth  to  a  rack. 
"     2         "  "          "        55     "       "  134  teeth. 

"     3         "  "  "        35     "       "     54 

"    4         "  "  "         26     "        "     34 

«     -         «  «  u        2I     «        «     2- 


J2 


PROVIDENCE,    R.    I. 


51 


52  BROWN    &    SHARPE   MFG.    CO. 

PART  II.  —  DETERMINE   THICKNESS  OF  CUTTER. 

It  is  very  evident  that  a  bevel  gear  cutter  cannot  be  thicker 
than  the  width  of  the  space  at  small  end  of  tooth  ;  the  practice 
is  to  make  cutter  .005"  thinner.  Theoretically  the  cutting  angle 
(h}  is  equal  to  pitch  angle  less  angle  of  bottom  (or  h  =  a  —  /?'). 
Practically,  however,  better  results  are  obtained  by  making 
h  =  a  —  ft  (substituting  angle  of  top  for  angle  of  bottom),  and 
in  calculating  the  depth  at  small  end,  to  add  the  full  clearance 
(/)  to  the  obtained  working  depth,  giving  equal  amount  of 
clearance  at  large  and  small  end.  This  is  done  to  obtain  .a 
tooth  thinner  at  the  top  and  more  curved.  As  the  small  end 
of  tooth  determines  the  thickness  of  cutter,  we  shall  have  to 
find  the  tooth  part  values  at  small  end.  From  the  diagram  it 
will  be  seen  that  the  values  at  large  end  are  to  those  at  small 
end  as  their  respective  apex  distances  (a  b  and  a  /).  The 
numerical  values  of  these  can  be  taken  from  the  diagram  and 
the  quotient  of  the  larger  in  the  smaller  is  the  constant  where- 
with to  multiply  the  tooth  values  at  large  end,  to  obtain  those 
at  small  end.  In  our  example  we  find  : 

ai  =  Jf  =  "655  =  constant  For  5  P  we  have  : 


J  =  .2000 

/=.Q3i4  7  =  ^314 

.23i4  /  +/=.i624 

D"  +  /=  .4314.  s' 


From  the  foregoing  it  is  evident  that  a  spur  gear  cutter 
could  not  be  used,  since  a  bevel  gear  cutter  must  be  thinner. 

If  in  gears  of  more  than  30  teeth  the  faces  are  proportion- 
ately long,  we  select  a  cutter  whose  curve  corresponds  to  the 
midway  section  of  the  tooth.  The  curve  of  the  cutter  is  found 
by  the  method  explained  in  Part  I.  of  this  Chapter. 


PROVIDENCE,    R.    I.  53 


IX. 


DIRECTIONS   FOR  CUTTING   BEVEL    GEARS 
WITH  ROTARY  CUTTER. 

(Fig.   20.) 

In  order  to  obtain  good  results,  the  gear  blanks  must  be  of 
the  right  size  and  form.     The  following  sizes  for  each  end  of 
the  tooth  must  be  given  the  workman  : 
Total  depth  of  tooth. 
Thickness  of  tooth  at  pitch  line. 
Height  of  tooth  above  pitch  line. 

These  sizes  are  obtained  as  explained  in  Chapter  VIII. 
The   workman  must  further  know  the  cutting  angle  (see 
(formula  on  page  13  and  compare  Chapter  VIII.),  and  be  pro- 
vided with  the  proper  tools  with  which  to  measure  teeth,  etc. 
In  cutting  a  gear  on  a  universal  milling  machine  the  opera- 
tions and  adjustments  of  the  machine  are  as  follows  : 

1.  Set  spiral  bed  to  zero  line. 

2.  Set  cutter  central  with  spiral  head  spindle. 

3.  Set  spiral  head  to  the  proper  cutting  angle. 

4.  Set  the  index  on  head  for  the  number  of  teeth  to  be  cut, 
leaving  the  sector  on  the  straight  or  numbered  row  of  holes, 
and  set  the  pointer  (or  in  some  machines  the  dial)  on  cross-feed 
screw  of  milling  machine  to  zero  line. 

5.  As  a  matter  of  precaution,  mark  the  depth  to  be  cut  for 
large  and  small  end  of  tooth  on  their  respective  places. 

6.  Cut  two  or  three  teeth  in  blank  to  conform  with  these 
marks  in  depth.     The  teeth  will  now  be  too  thick  on  both  their 
pitch  circles. 

7.  Set  the  cutter  off  the  center  by  moving  the  saddle  to  or 
from  the  frame  of  the  machine  by  means  of   the  cross-feed 
screw,  measuring  the  advance  on   dial  of  same.     The  saddle 
must   not  be    moved    further  than  what   to   good   judgment 


54 


BROWN    &    SHARPE    MFG.    CO. 


PROVIDENCE,    R.    I.  55 

appears  as  not  excessive  ;  at  the  same  time  bearing  in   mind 
that  an  equal  amount  of  stock  is  to  be  taken  off  each  side  of 

tooth. 

• 

8.  Rotate  the  gear  in  the  opposite  direction  from  which  the 
saddle  is  moved  off  the  center,  and  trim  the  sides  of  teeth  (A) 
(Fig.  20.) 

9.  Then  move  the  saddle  the  same  distance  on  the  opposite 
side  of  center  and  rotate  the  gear  an  equal  amount  in  the 
opposite  direction  and  trim  the  other  sides  of  teeth  (C). 

10.  If  the  teeth  are  still  too  thick  at  large  end  E,  move  the 
saddle  further  off  the  center  and  repeat  the  operation,  bearing 
in  mind  that  the  gear  must  be  rotated  and  the  saddle  moved 
an  equal  amount  each  way  from  their  respective  zero  settings. 

It  is  generally  necessary  to  file  the  sides  of  teeth  above  the 
pitch  line  more  or  less  on  the  small  ends  of  teeth,  as  indicated 
by  dotted  lines  F  F.  This  applies  to  pinions  of  less  than  30 
teeth. 

For  gears  of  coarser  pitch  than  5  diametral  it  is  best  to 
make  one  cut  around  before  attempting  to  obtain  the  tooth 
thickness. 

The  formulas  for  obtaining  the  dimensions  and  angles  of 
gear  blanks  are  given  in  Chapter  III. 


BROWN    &    SHARPE    MFG.    CO. 


THE    INDEXING  OF  ANY  WHOLE  OR    FRAC- 
TIONAL  NUMBER. 

(Fig.   21.) 


< — Change  Gear 

Fig.  21. 


In  indexing  on  a  machine  the  question  simply  is  :  How 
many  divisions  of  the  machine  index  have  to  be  advanced  to 
advance  a  unit  division  of  the  number  required.  To  which 
is  the 

divisions  of  machine  index 


answer 


number  to  be  indexed 


Suppose  the  number  of  divisions  in  index  wheel  of  machine 
to  be  216. 


EXAMPLE  I. — Index  72. 


Answer  : 


2X6 

72 


(3  turns  of  worm). 


PROVIDENCE,    R.    I.  57 

EXAMPLE  II.  —  Index  123. 

^=i+-93 
123  123 

If  now  we  should  put  on  worm  shaft  a  change  gear  having 
123  teeth,  give  the  worm  shaft,  Fig.  21,  one  turn,  and  in  addi- 
tion thereto  advance  93  teeth  of  the  change  gear  (to  give  the 
fractional  turn),  we  would  have  indexed  correctly  one  unit  of 
the  given  number,  and  so  solved  the  problem.  Should  we  not 
have  change  gear  123  we  may  try  those  on  hand.  The  ques- 
tion then  is  :  How  many  teeth  (x)  of  the  gear  on  hand  (for 
instance  82)  must  we  advance  to  obtain  a  result  equal  to  the 
one  when  advancing  93  teeth  of  the  123  tooth  gear  ?  We  have  : 

.93_  =  X_  where  ^  =  62 
123      82 

EXAMPLE  III.  —  Index  365,  change  gear  147. 

£!<?  =  JL  where  j  =  87  -  -1_ 

365      U7  365 

Here  147  is  the  change  gear  on  hand.  In  indexing  for  a  unit 
of  365  we  advance  STteeth  of  our  147  tooth  gear.  It  is  evident 
that  in  so  doing  we  advance  too  fast  and  will  have  indexed 
three  teeth  of  our  change  gear  too  many  when  the  circle  is 
completed.  To  avoid  having  this  error  show  in  its  total  amount 
between  the  last  and  the  first  division,  we  can  distribute  the 
error  by  dropping  one  tooth  at  a  time  at  three  even  intervals. 

EXAMPLE  IV.  —  Index  190. 

216  =  T  +   26 

I9o  190          Change  gear  on  hand  90  T 

26        x  6^ 

—  =  —  where  x  —  I2  +  - 
190      90  190 

To  distribute  the  error  in  this  case  we  advance  one  addi- 
tional tooth  at  a  time  of  the  change  gear  at  six  even  intervals. 

EXAMPLE  V.  —  Index  117.3913. 

216  986087 

~ 


11739*3 

This  example  is  in  nowise  different  from  the  preceding 
ones,  except  that  the  fraction  is  expressed  in  large  numbers. 
This  fraction  we  can  reduce  to  lower  approximate  values, 
which  for  practical  purposes  are  accurate  enough.  This  is 
done  by  the  method  of  continued  fractions.  [For  an  explana- 


58  BROWN    &    SHARPE    MFG.    CO. 

tion  of  this  method  we  refer  to  our  "  Practical  Treatise  on 
Gearing."] 

986087 
"73913 

986087)  1173913  (i 
986087 

^87826)  986087  (5 
939130 

46957)  187826  (3 
140871 

46955)  46957  d 
46955 

2)  46955  (23477 
46954 

1)2(2 
2 


i  +  i 


"73913 


23477  +  £ 

2 


5  <r=3        i       23477 


ar=jt      b  =  _5      d=  16     21     493033      986087 
<xl  =  i     bl  —  6     a?1  =  19     25     586944     1173913 

NOTE. — Find  the  first  two  fractions  by  reduction       =  -  and  — : —  =  z  ;     the 

i        i          i  +  i^      6 

5 

others  are  then  found  by  the  rule  \  b  c  +  a  ~  d 

1  dl  c  +  a1  =dl 

The  fraction  \\  is  a  good  approximation;  putting  therefore 
a  change  gear  of  25  teeth  on  worm  shaft,  we  advance  (beside 
the  one  full  turn)  21  teeth  to  index  our  unit. 

Of  course,  in  using  any  but  the  correct  fraction  we  have  an 
error  every  time  we  index  a  division  ;  so  that  when  indexed 
around  the  whole  circle,  we  have  multiplied  this  error  by  the 
number  of  divisions. 

In  the  present  example  this  error  is  evidently  equal  to  the 
difference  between  the  correct  and  the  approximate  fraction 
used.  Reducing  both  common  fractions  to  decimal  fractions 
we  have  : 

986o87   =  .84000006 
I3 

'—  =     84.OOOOOO  •  i        i  •      •     • 

2C-      '—* =  error  in  each  division. 

5       .00000006 


PROVIDENCE,    R.    I.  59 

.00000006  *  117.3913  =  .00000703348  total  error  in  complete 
circle.  This  error  is  expressed  in  parts  of  a  unit  division.  (To 
find  this  error  expressed  in  inches,  multiply  it  by  the  distance 
between  two  divisions,  measured  on  the  circle.)  In  this  case 
the  approximate  fraction  being  smaller  than  the  correct  one, 
in  indexing  the  whole  circle  we  fall  short  .00000703348  of  a 
division. 

EXAMPLE  VI. — Index  15.708 
216 


I 

3 

4 

196 

983 

I 

261 

1309 

In  using  the  approximation  -j-f  J  tne  error  f°r  eacn  division 
(found  as  above)  will  be  .000002917,  for  the  whole  circle 
.0000458.  In  this  case,  the  approximation  being  larger  than 
the  correct  fraction,  we  overreach  the  circle  by  the  error. 


6o 


BROWN    &    SHARPE    MFG.    CO. 


XI. 


THE  GEARING  OF   LATHES   FOR   SCREW 
CUTTING. 

(Figs.  22,  23.) 

The  problem  of  cutting  a  screw  on  a  lathe  resolves  itself  into 
connecting  the  lathe  spindle  with  the  lead  screw  by  a  train  of 
gears  in  such  a  manner  that  the  carriage  (which  is  actuated  by 


Simple  Gearing. 

Fig.  22. 


PROVIDENCE,    R.    I. 


6l 


the  lead  screw)  advances  just  one  inch,  or  some  definite  dis- 
tance, while  the  lathe  spindle  makes  a  number  of  revolutions 
equal  to  the  number  of  threads  to  be  cut  per  inch. 

The  lead  screw  has,  with  the  exception  of  a  very  few  cases, 
always  a  single  thread,  and  to  advance  the  carriage  one  inch  it 
therefore  makes  a  number  of  revolutions  equal  to  its  number 


Compound  Gearing 

Fig.  23. 


of  threads  per  inch.  Should  the  lead  screw  have  double 
thread,  it  will,  to  accomplish  the  same  result,  make  a  number 
of  revolutions  equal  to  half  its  number  of  threads  per  inch.  It 
follows  that  we  must  know  in  the  first  place  the  number  of 
threads  per  inch  on  lead  screw. 


62  BROWN    &    SHARPE    MFG.    CO. 

It  ought  to  be  clearly  understood  that  one  or  more  inter- 
mediate gears,  which  simply  transmit  the  motion  received  from 
one  gear  to  another,  in  no  wise  alter  the  ultimate  ratio  of  a 
train  of  gearing.  An  even  number  of  intermediate  gears 
simply  change  the  direction  of  rotation,  an  odd  number  do  not 
alter  it. 

The  gearing  of  a  lathe  to  solve  a  problem  in  screw  cutting 
can  be  accomplished  by 

A.  Simple  gearing. 

B.  Compound  gearing. 

Referring  to  the  diagrams,  Figs.  22  and  23,  we  have  in  Fig. 
22  a  case  of  simple,  and  in  Fig.  23  a  case  of  compound  gear- 
ing. 

In  simple  gearing  the  motion  from  gear  E  is  transmitted 
either  directly  to  gear  Ron  lead  screw  or  through  the  interme- 
diate F.  In  compound  gearing  the  motion  of  E  is  transmitted 
through  two  gears  (G  and  H)  keyed  together,  revolving  on  the 
same  stud  #,  by  which  we  can  change  the  velocity  ratio  of  the 
motion  while  transmitting  it  from  E  to  R.  With  these  four 
variables  E,  G,  H,  R,  we  are  enabled  to  have  a  wider  range  of 
changes  than  in  simple  gearing. 

B  and  C,  being  intermediate  gears,  are  not  to  be  considered. 
If,  as  is  generally  the  case,  gear  A  equals  gear  D,  we  disregard 
them  both,  simply  remembering  that  gear  E  (being  fast  on 
same  shaft  with  13)  makes  as  many  revolutions  as  the  spindle. 
Sometimes  gear  D  is  twice  as  large  as  gear  A,  then,  still  con- 
sidering gear  E  as  making  as  many  revolutions  as  the  spindle, 
we  deal  with  the  lead  screw  as  having  twice  as  many  threads 
per  inch  as  it  measures. 


SIMPLE   GEARING. 

Let  there  be :    the  number  of  teeth  in  the  different  gears 
expressed  by  their  respective  letters,  as  per  Fig.  22,  and 

s  =  threads  per  inch  to  be  cut, 
L  =  threads  per  inch  on  lead  screw  ;  then 

i.  *_R 

L~D 


PROVIDENCE,    R.    I.  63 

If  now  one  of  the  two  gears  D  and  R  is  selected,  the  other 
will  be  : 

R       sD        n      LR 
R=  -T^         D  .— 

L  s 

2.  The  two  gears  may  be  found  by  making 

—  ^  T   r  where/  may  be  any  number. 

LJ    p    ]L-I  y 

3.  The  above  holds  good  when  a  fractional  thread  is  to  be 
cut,  but  if  the  fraction  is  expressed  in  large  numbers,  as,  for 
instance,  s  =  2.833  (2-$^),  we  first  reduce  this  fraction  (T8^3A)  to 
lower  approximate  values  by  the  process  of  continued  fraction 
(see  pages  57  and  58). 

833)  icoo  (i 

833 

167)  833  (4 


165) 

167(1 

165 

2)  l65  (82 

16 

5 

4 

1)2(2 

2 

O 

I 

4 

I                  82 

2 

I 

4 

5        4H 

833 

I 

5 

•6         497 

IOOO 

L  =  .833  (nearly)  and  s  =  2$- 
u  6 

If  in  this  case  L  =  4,  and  we  select  D  =  48,  then,  since 
R  =  *»     R  =  34 

" 

COMPOUND  GEARING. 

4.  In  a  lathe  geared  compound  for  cutting  a  screw  the 
product  of  the  drivers  (E  and  H,  Fig.  23)  multiplied  by  the  num- 
ber of  threads  to  be  cut  must  equal  the  product  of  the  driven 
(G  and  R)  multiplied  by  the  number  of  threads  on  lead  screw. 
This  is  expressed  by 

E.H.;=G.R.Lor  JL_M_£  =  i 

(jr.  K  .  L, 


64  BROWN    &    SHARPE    MFG.    CO. 

If  three  of  the  gears  E,  H,  G,  R  have  been  selected,  the 
fourth  one  would  be  either 


or 


or 


or 


GL 


_RGL  =L  /  R.G  \ 
EH  VL.E.H/ 


If  a  fractional  thread  is  to  be  cut,  as  under  "3,"  we  reduce 
the  fraction  to  lower  approximate  values. 

EXAMPLE. — Gear  for  5.2327  threads  per  inch,  lead  screw  is 
6  threads. 

_   2327 

IOOOO 

2327)  looco  (4 
9308 

692)  2327  (3 
2076 

~25l)  692  (2 
502 

190)  251  (I 
190 

61)  190  (3 
183 
7)  61  (8 

~5)7(i 
5_ 

2)5(2 

4 

1)2(2 

2 


i      3       7      10      37       306       343       992       2327 


4     13     30     43     159     1315      1474     4263     loooo 

—  =  .2327  (nearly)  and  5.2327  =  5— 
43  43 

Selecting  E  =  43,  H  =  52,  R  =  50,  and 

•5tt- 


PROVIDENCE,    R.    I.  65 

5.  The  examples  so  far  given  all  deal  with  single  thread. 
The  pitch  of  a  screw  is  the  distance  from  center  of  one  thread  to 
the  center  of  the  next.  The  lead  of  a  screw  is  the  advance  for 
each  complete  revolution.  In  a  single  thread  screw  the  pitch 
is  equal  to  the  lead,  while  in  a  double  thread  screw  the  pitch 
is  equal  to  one-half  the  lead  ;  in  a  triple  thread  screw  equal  to 
one-third  the  lead,  etc. 

If  we  have  to  gear  a  lathe  for  a  many-threaded  screw 
(double,  triple,  quadruple,  etc.),  we  simply  ascertain  the  lead, 
and  deal  with  the  lead  as  we  would  with  the  pitch  in  a  single 
thread  screw,  /.  e.,  we  divide  one  inch  by  it,  to  obtain  the  num- 
ber of  threads  for  which  we  have  to  gear  our  lathe. 

EXAMPLE.  —  Gear  for  double  thread  screw,  lead  =  .4654. 
Number  of  threads  per  inch  to  be  geared  for  is  : 

1      =  _!_:=  2.1487 
Lead       -4654 

Lead  screw  is  four  threads  per  inch. 

As  in  previous  examples,  we  reduce  the  fraction  .1487=^^^5- 
to  lower  approximate  values  by  the  process  of  continued  frac- 
tion. 

From  the  different  values  received  in  the  usual  way  we 
select  : 

\l  =  .1487  (nearly)  and  2.1487  =  2^ 

We  have  therefore  : 


Selecting     •<  G  =  30 
(  H=  40 

R  _  E  .  H  .  s  _  74  .  40  .  2ft  _ 
G  .  L  30  .  4 

NOTE.  —  In  using  any  but  the  original  fraction  we  commit  an  error.  This  error 
can  be  found  by  reducing  the  approximate  fraction  used  to  a  decimal  fraction,  and 
comparing  it  with  the  original  fraction.  In  the  above  example  the  original  fraction  is 

.1487    and 
H  =  .  14864 
Error  =  .00006  inch  in  lead. 


In  cutting  a  multiple  screw,  after  having  cut  one 
thread,  the  question  arises  how  to  move  the  thread  tool  the 
correct  amount  for  cutting  the  next  thread. 


66  BROWN    &    SHARPE   MFG.    CO. 

In  cutting  double,  triple,  etc.,  threads,  if  in  simple  or  com- 
pound gearing  the  number  of  teeth  in  gear  E  is  divisible  by 
2,  3,  etc.,  we  so  divide  the  teeth  ;  then  leaving  the  carriage 
at  rest  we  bring  gear  E  out  of  mesh  and  move  it  forward  one 
division,  whereby  the  spindle  will  assume  the  correct  position. 

Is  E  not  divisible  we  find  how  many  teeth  (V)  of  gear  R 
are  advanced  to  each  full  turn  of  the  spindle.  Dividing  this 
number  by  2  for  double,  by  3  for  triple  thread,  etc.,  we 
advance  R  so  many  teeth,  being  careful  to  leave  the  spindle  at 
rest. 

We  have  for  simple  gearing  : 


V-  E 


for  compound  gearing  : 


G.R 

If  in  simple  gearing  both  E  and  R  are  not  divisible,  one 
remedy  would  be  to  gear  the  lathe  compound  ;  or  the  face- 
plate may  be  accurately  divided  in  two,  three  or  more  slots, 
and  all  that  is  then  necessary  is  to  move  the  dog  from  one  slot 
to  another,  the  carriage  remaining  stationary. 


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